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Huntsville City Schools Real Talk Mentoring Program Needs You!
It's time for some Real Talk. "It takes a village to raise a child, and it takes a community to raise a school." During her many years of working in the school system, Helen Drake is adamant that students need face-to-face contact with supporters in the community. "Some of these students don't have parents. We need community members to step up and say, 'I care.' That's what I want the students to see."
Helen Drake, Student Support Specialist for Huntsville City Schools, started the HCS Community School-Based Mentor program, called Real Talk, in January 2018. The program provides a structured way for students to connect with people in the community – during lunch. Facilitated by the principals, teachers, and counselors "Power Hour" helps students engage with different youth organizations, churches, businesses, or clubs during their lunch break.
"The first step: let the students know that they have a community that believes in them, and supports them in academics," says Drake.
Raising Up Our Students
Drake describes one conversation with a student that helped develop the pilot program, which ran from January 2018 – May 2018.
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"You were probably chosen to be part of the pilot program because you have a kind heart and like to help others," I looked at the young man and said, "Do you like to lead and help others solve problems?"
A look of pride came over his face and he said, "Yes, I do."
I told him, "You were chosen for that leadership ability and because you influence others. That's why we need your help to get this program started."
The program provides students with meaningful time with a mentor that wants to help students thrive in school. Huntsville High School, Grissom High School, Lee High, New Century High, Mae Jemison High, and Columbia High each had 20 students that participated in the program. Students are encouraged to learn how to serve and support their classmates and others, as they gain academic and leadership skills.
Why Real Talk Works
What makes the program unique is the resources available to mentors.
Mentors are provided with some of the same professional development training that is offered to school staff. "By training our mentors, we make the support base stronger," Drake says. Kenny Anderson, City of Huntsville Multicultural Affairs Officer, has partnered with Drake to facilitate mentor training as well as post mentor training online. The added convenience makes it possible for people to receive training when it works best for their schedule.
Spring Break Camps in Huntsville
Real Talk Needs People with a Heart for Children
Families, community groups, businesses, and churches are all welcome to join the program, become a speaker, or donate to the program. Whether a speaker, mentor or a donor, often the community members learn something from the students.
To learn more about how to become a Real Talk mentor, a speaker, or donate, contact:
Helen Drake
Student Support Specialist
Email Helen or call 256-428-6888 ext. 4
You can also sign up to mentor online.
This article first appeared on the Huntsville City Blog.
[themify_box color="lavender"]ABOUT THE AUTHOR: Analyn Hardy is one of the Digital Media Specialists for the City of Huntsville. Her work includes data and analytics for all things that involve a screen including social media, website, and e-newsletters. Read more of her hyperlocal work here.[/themify_box]
Mentoring Program Aims to Connect Students to Community Members
HCS Virtual Volunteer Center
How Parents Can Best Help Their Teachers Help Their Kids
As a hyper-local website focused on all aspects of parenting in and around Huntsville, AL, and the Tennessee Valley, Rocket City Mom occasionally asks local parents to submit their stories for publication. This is part of our continual effort to represent varied viewpoints and experiences on our site. However, these articles should not be seen as necessarily expressing the views of Rocket City Mom Media Group, LLC.
Guest Contributor Oct 11
New Year Resolutions That Have Nothing To Do With Your Diet, Fitness, or Finances
Great Ways to Get Fit & Healthy in Huntsville
Mythbusting Elderberry Syrup, Essential Oils, & Alternative Remedies for Kids
Where Kids Eat Free Today
Where to Find Story Times Today
Beaks and Barks
Castles & Dragons Day Camp at Alexander's Martial Arts
Alexander's Martial Arts – Madison
The Matrix Whiz Kids Open Gym (Under 6)
Matrix Gym
Daily Free Fun
Subscribe to 365 Days of Free
Teal Pumpkin Project About Inclusion & Fun
Screen Time Solutions: The Coin System
How Parents Can Best Help Teachers Help Their Kids
How Parents Can Help Improve Their School's Report Card
Great Big Back to School Giveaway
5 Ways to Go Back to School Shopping in Huntsville & Madison County
5 "Be's" to Prepare Your Child for Tomorrow, Today
Superintendent Nominees for Huntsville City and Madison City are Interviewed


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from src.util.task import Task
import src.util.task as task

def testNetStringConversion():
    vertices = [1, 0, 1 , 1] # which vertices are present in subgraph
    edges = [1, 3 , 4] # index of edges present in subgraph
    bloomHash = 'hash1'
    serverHash = 'hash2'
    expectedString = "1011$3$1$3$4$hash1$hash2"
    t = Task(vertices, edges, bloomHash, serverHash)
    assert t.toNetString() == expectedString
    print "To Net String Conversion Passed"

def testObjectConversion():
    netString = "1011$3$1$3$4$hash1$hash2"
    t = task.toTaskFromNetString(netString)
    assert t.vertices == [1, 0, 1, 1]
    assert t.edges == [1, 3, 4]
    assert t.bloomHash == 'hash1'
    assert t.serverHash == 'hash2'
    assert t.toNetString() == netString
    print "To Object Conversion Passed"

if __name__ == '__main__':
    testNetStringConversion()
    testObjectConversion()

redpajama_set_name: RedPajamaGithub


text:
LIVERPOOL CAN THRIVE WITHOUT UNESCO WORLD HERITAGE STATUS
Losing its UNESCO world heritage status offers Liverpool an opportunity to better plan how major infrastructure is interlinked throughout the city region, argues Keith York of Civic Engineers.
News that UNESCO has withdrawn Liverpool's world heritage site status has been met with a fairly mixed response in the business and political worlds.
From my point of view, Liverpool can be better off without its UNESCO title and can continue to thrive and prosper without it.
Other than emboldening the city with a degree of prestige in certain circles, the status itself is not responsible for driving much of the positive change which Liverpool has achieved over the past 17 years since it was first announced.
In fact, one of the biggest drivers of that change has been something that the world heritage status has threatened to hold back – better infrastructure. It is our view that such an extent of the city's waterfront should never have fallen within the UNESCO listing. In doing so, it risked sterilising derelict and unremarkable areas of docklands, particularly along the city's North Shore.
Instead, Liverpool City Council and its allies have had the courage to embrace progress and change, whilst also investing millions in protecting the areas with undoubted historic value, such as the three graces and the Royal Albert Dock.
There is no doubt that crowds of people will continue to visit Liverpool and the wider city region because of its unrivalled heritage and its historic architecture. But contemporary developments such as Liverpool One, The Museum of Liverpool, and Liverpool Waters, have also become significant and necessary parts of Liverpool's modern story, ensuring that the city remains an attractive prospect for those choosing to live, work and invest here.
Cities are intrinsically changing places that evolve over centuries, with social, political, and industrial networks converging across any urban environment that is worth its salt. Having greater capacity for height and density is a positive and an essential feature of a modern urban settlement.
Of course, in cities like Liverpool where strength of heritage is a key driving force for decision-making, new plans and developments must also be tempered with appropriateness and contextual consideration. While UNESCO status prompted such considerations, it also placed too many limitations on an environment where plenty of other checks and balances are already in place.
The future development of Liverpool needs to continue to be assertive and confident. People will still come to Liverpool and see the city and its famous waterfront for themselves, something which many UNESCO panel members have failed to do even once since the conferring of the city's status back in 2004.
The work that Civic Engineers has delivered at Liverpool Lime Street Gateway is an example of sensitively positioning the new with the old, restoring the listed frontage of the station right on the doorstep of St George's Hall.
As development projects like this continue at pace across Liverpool and now without the limitations of UNESCO, there is an opportunity to better plan how major infrastructure is interlinked throughout the city-region, ensuring for example that Liverpool has a strong focus upon active travel combined with public transport use.
Liverpool will continue to grow and adapt whilst protecting its history for all the world to admire and those who see the removal of the UNESCO title as a disappointment in terms of the city's prospects would be wise to reframe this news as an opportunity for Liverpool to progress sensitively and appropriately for its 21st century needs.
Keith York is an associate at Civic Engineers.

redpajama_set_name: RedPajamaCommonCrawl


text:
Los Angeles Misioneros es un equipo de fútbol de los Estados Unidos que juega en la USL Premier Development League, la cuarta liga de fútbol más importante del país.

Historia
Fue fundado en el año 2006 en la ciudad de Los Ángeles, California con el nombre Los Angeles Storm tras haber adquirido los derechos del Springfield Storm y su primer entrenador fue Chris Volk.

Su primer partido fue un empate 1-1 ante el San Fernando Valley Quakes en una temporada en la que terminaron en quinto lugar de su división.

En el año 2008 cambian su nombre por el de Los Angeles Legends tras fusionarse con dos clubes juveniles, ganaron el primer partido de esa temporada 1-0 al Orange County Blue Star y clasificaron por primera vez a la US Open Cup, aunque fueron eliminados en primera ronda por el Christal Palace Baltimore de la USL Second Division.

Al año siguiente el club fue adquirido por los propietarios del Los Angeles Blues y su nombre lo cambiaron a Los Angeles Azul Legends y su nuevo logo lo mostraron al público en abril del 2010. Entre 2010 y 2011 cambiaron su nombre por el que tienen actualmente, el cual fue ratificado en enero del 2011.

Estadios
 Citrus Stadium; Glendora, California (2006–2007)
 Cougar Stadium; Azusa, California (2008)
 Keegan Stadium en Bishop Mora Salesian High School; East Los Angeles, California (2009–2010)
 Ayala Stadium; Chino, California 1 juego (2009)
 Crenshaw High School Field; Crenshaw, California 4 juegos (2010–2011)
 Morningside High School Ground; Inglewood, California (2011)
 George Washington Carver Middle School; Los Ángeles, California (2012)
 California State University Field; San Bernardino, California (2013–)
 Glendale Sports Complex; Huntington Park-Glendora-Glendale, California (2014–)

Presidentes
  Freddy Avellaneda (Administrador) (2011-)
  José Antonio Jiménez (2012-)
  Daniel Collazo (2014-)

Entrenadores
  Chris Volk (2006–2007)
  Phil Wolf (2008)
  Brent Whitfield (2009)
  Salvador Moran (2010)
  Lloyd Kinnear (2010)
  Marlon Iván León (2011–2012)
  Ever Morataya (2012–)
  Cruz Hernández (2014–)

Jugadores

Jugadores destacados

  Joe Barton
  Tony Beltran
  Alex Bengard
  Kraig Chiles
  Gabriel Farfan
  Michael Farfan
  Lance Friesz
  Erlys García
  Maxwell Griffin
  Lester Moré
  Kyle Nakazawa
  Raul Palomares

  Davis Paul
  Mike Randolph
  Eders Roldan
  Ross Schunk
  Erasmo Solorzano
  Brent Whitfield
  Gerson Mayen
  Víctor Sánchez
  Edwin Miranda
  Vincent Weijl
  Mohammed Takiyudin
  Jonathyn Lomeli

Equipo 2014

Referencias

Enlaces externos
 Página oficial (en inglés)
 PDL - Perfil (en inglés)
 Los Angeles Legends - Club Asociado (en inglés)

Equipos de la USL League Two
Equipos de fútbol fundados en 2006
Clubes y equipos deportivos de Los Ángeles
Equipos de fútbol que se han fusionado y fusiones

redpajama_set_name: RedPajamaWikipedia


text:
Untold Secret to Melodic Bass lesson: Anchors, Pivots and Lead-Ins
by Jon Burr Monday, June 15th, 2009
This is the first chapter of "The Untold Secret to Melodic Bass." Be sure to check out Jon's previous lesson, Melodic Bass Lines, as well as the complete The Untold Secret to Melodic Bass in downloadable PDF or Kindle Book formats.
I hope you, the bassist-reader, find the material useful; it's an approach to the construction of bass lines based on my years as a working bassist, taken from observation, trial-and-error experience, conversations with bassists and other musicians, and anecdotes of oral tradition that I was privileged to hear from my peers, idols and mentors. Your comments, observations, complaints, criticisms, insights or endorsements are welcome and I'll make every effort to respond!
The secret to confidence in playing the bass—or in anything, for that matter—is knowing what you're doing. When you're playing the bass, having knowledge of the harmony—the chord changes—is not only vital, it's inescapable. You have to know the chord changes. This book assumes that you know the changes, or have developed the ability to figure them out quickly. Learning chord changes—and tunes—is a great subject, worthy of a whole other book, but this book is about what to play on them; how to create bass lines, melodies, or patterns in any style. We'll learn how to think like a composer of bass lines; how to plan ahead, and make choices that are appropriate to the style of the song.
Most music we'll play has an underlying structure—a "form"—whether it's a 12-bar Blues, a 32-bar standard in AABA form (or 16&16), or whatever else it might be, depending on what the composer wrote. Within that form, the harmony progresses with its "harmonic rhythm," meaning that the chords happen at certain times every time we play through the form of the song. The underlying rhythm is very important to the choices that we make as well; the rhythmic environment determines the "style," whether it's swing, 2-beat, cha-cha, bossa, samba, merengue, waltz, shuffle, or any of countless variants that fall under the heading of "contemporary," whether it's rock, funk, hip-hop, house, new age, alternative, or whatever else.
So, what notes do we play? How do we make interesting, melodic lines, enhance the music, yet at the same time "follow the rules"—which basically means play for the music, make what we play fit the music and serve the music and our fellow players—and still be able to "make a statement," put our stamp on the music? How can we assert a musical identity, be creative, and yet "take care of business" at the same time?
By using the hidden power of the structure of harmony and rhythm!
The root of the chord is the strongest sound we can make on the bass, and essential to our role; the notes of the chords contain a property that we could describe as "harmonic magnetism." We are going to explore these "magnetic points" with a primary focus on "anchors" and "pivots." Once we cover these points of responsibility, we are free to create and embellish, using "lead-ins." As we develop these melodies, they can be enhanced further by the incorporation of dynamics; volume rises and falls with the rise and fall of tension and release.
The bass is primarily responsible to provide anchor notes at certain points in the music. Simply put: we are generally responsible to play a root in a low register when a harmony first occurs; doing so underscores to the band "this is the chord!" If we cover the basic areas of our responsibility as bass players—which involve "showing up" at definable points in the music with the right information—then we have considerable freedom in how we get to the next "signpost" or anchor point.
A "pivot" is similar to an anchor in that it communicates fundamental harmony at an expected time; however, it is most often not the root of the chord, and not at the beginning of the harmonic instance; the most common pivot is the fifth of the chord—but it can be any chord tone, depending on taste and circumstances; and its most common location in the rhythm is on the 3rd beat, assuming 4/4—or, rather, halfway through the bar. There are many popular songs built around a specific bass line; in these songs, the pivot contained in the fixed line is part of the bassist's responsibility, but development can still occur within these kinds of frameworks. We will explore this as the book progresses.
Lead-Ins
Lead-Ins are the connecting material…these are the bassist's creative opportunity. Once the bassist has delivered an anchor, he can employ lead-ins of a wide variety to approach the next benchmark—the pivot—and then transition from the pivot to the next anchor. Lead-ins come in all shapes and sizes, rhythmic and melodic configurations, and they are the bassist's playgound.
As is the case in every set of rules, rules are made to be broken. It is not always the case that the bass has to play an anchor on the first beat of every chord change. As development progresses, anchors and pivots can be rhythmically displaced. Sometimes, in walking bass lines, the distances between anchors can increase; lead-ins can occur on strong rhythmic beats, pivots can be used as anchors, and lead-ins can be extended. We will talk about all of these options as the book progresses.
Internal Dynamics
We are also going to cover the subject of internal rhythmic dynamics, which we would define as dynamic variation within a particular musical phrase. Tension and release is as important in rhythm as it is in melody, and the foundation of rhythmic music of all kinds is accents, where they fall, and how much emphasis they get. Bassists need to know something about the traditions that exist here, which seem to have been largely overlooked in written pedagogy, but have been transmitted orally since the first strike of a drum.
Rhythmic Overlays
"Rhythmic Overlays" are defined patterns of accents superimposed on the rhythmic substrate, derived from numeric patterns or other material such as the rhythm of lyrics. In the section on overlays, below, we will discuss some to the sources for these overlays.
Harmonic Dynamics
This is a term we haven't seen before in our musical studies; it shows up in biophysics and chemistry referring to something else entirely, but this term will be used to describe relationships between musical harmonic events such as cadences (and their resolution) and the volume level or intensity with which they are played. We can define "harmonic dynamics" as "changes of volume and intensity suggested by the progression of the chords through tension and resolution."
…to be continued!
In case you want to see more right now…
The Untold Secret to Melodic Bass is available as a pdf download or as a Amazon Kindle book.
Related Topics: bass lessons, jon burr
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Sourcing for the right raw materials are as critical as buying the right home pellet maker from us. At RAW, we are committed to working with only the companies that sell recycled sawdust waste as raw material.
Each time a tree is milled into a log, the sawdust generated can be recycled. It does not need to be disposed as waste. And that's where you and I come in – to change that.
When you place an order to purchase recycled sawdust, you contribute to the reduction of cutting additional trees to produce more sawdust.
Sawdust waste that is left unattended will decompose and emit carbon – so using it to make wood pellets will reduce the amount of carbon emitted.
Last but not least, when you purchase a home pellet maker from us, we will help you to purchase recycled sawdust within your country. Although this isn't exactly an environmental reason, it does help to support your local industries.


Showing results by author "Emily Rubin"
When Stalina Folskaya flees her native Russia after the fall of the Soviet Union in 1991, she sets out for America with her dreams - and her painful past - firmly in tow. In St. Petersburg, Stalina was a trained chemist, but in America, disillusioned with her profession, she takes a job as a maid at the Liberty, a seedy, "short-stay" motel on the outskirts of Hartford, Connecticut. Even here, gutsy Stalina sees opportunity, convincing the owner to let her transform the motel into a destination with fantasy settings.


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Lack of regeneration and climatic vulnerability to fire of Scots pine may induce vegetation shifts at the southern edge of its distribution
Vilà-Cabrera A., Rodrigo A., Martínez-Vilalta J., Retana J. (2012) Lack of regeneration and climatic vulnerability to fire of Scots pine may induce vegetation shifts at the southern edge of its distribution. Journal of Biogeography. 39: 488-496.
Aim Forest ecosystems dominated by fire-sensitive species could suffer shifts in composition under altered crown fire regimes mediated by climate change. The aims of this study were to: (1) study the spatio-temporal patterns and the climatic distribution of fires in Scots pine (Pinus sylvestris) forests during the last 31years in north-eastern Spain, (2) evaluate the climatic vulnerability to fire of these forests in Spain, (3) analyse the regeneration of Scots pine after fire, and (4) predict the mid-term maintenance or replacement of Scots pine in burned areas. Location Catalonia (north-eastern Spain): the southern distribution limit of Scots pine. Methods We characterized the spatio-temporal and the climatic distribution of fires that occurred in Catalonia between 1979 and 2009. We used a generalized linear model to characterize the climatic vulnerability to fire of Scots pine in the whole of Spain. We assessed the regeneration of the species after crown fires in nine burned areas in Catalonia. The resulting data were integrated into a stochastic matrix model to predict the mid-term maintenance or replacement of Scots pine in burned areas. Results During the last three decades, Scots pine forests distributed in dry sites were most affected by fire. Our assessment of the vulnerability to fire of Scots pine forests in Spain as a whole, based on climatic and topographical variables, showed that 32% of these forests are vulnerable to fire, and that this proportion could increase to 66% under a conservative climate change scenario. Field data showed almost no regeneration of Scots pine after crown fires, and a limited capacity to recolonize from unburned edges, even in relatively old fires, with 90% of recruits located in the first 25m from the edge. This process could be delayed by the elapsed time for new recruits to achieve reproductive maturity, which we estimated to be c.15years. Finally, our matrix model predicted the replacement of burned Scots pine forests by oak (Quercus sp.) forests, shrublands or mixed resprouter forests. Main conclusions Increased vulnerability to fire of Scots pine forests under future, warmer conditions may result in vegetation shifts at the southern edge of the distribution of the species. © 2011 Blackwell Publishing Ltd.
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When wildfires engulfed Bastrop County, nearly 1600 families lost their homes and many other properties were significantly damaged. Homeowners and business owners that have insurance to cover damage to their property may be eligible for financial compensation from their insurance provider to help with the cost of replacement or repairs.
Unfortunately for some fire damage victims, the appraisers for insurance companies may underestimate the cost of repairs to property or may not offer you fair value for your claim. With the help of a legal professional, you and your loved ones may be able to avoid unnecessary complications and may be able to recover the funds you are rightly owed according to your policy. Contact the Bastrop wildfire insurance lawyers of Williams Hart at 800-639-8460 to discuss your options today.
When an insurance company refuses to stick to their agreement and honor the policies established by their customers, legal action may be necessary.
At Williams Hart, our Bastrop fire claim attorneys are ready to help you recover the funds you are owed to make your home or business property useable again. Contact our offices at 800-639-8460 to speak to a knowledgeable lawyer about your options.


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Calgary·FOOD AND THE CITY
Dandy brings 'fine dining drinking food' to Ramsay with new brewery and tasting room
One of Calgary's first nano-breweries, the Dandy Brewing Company, has outgrown its tiny industrial taproom and has made the move to Ramsay, taking over a space on 11th Street that was previously a motorcycle shop and garage.
Nano-brewery a welcome addition to the neighbourhood's burgeoning beer scene
Julie Van Rosendaal · for CBC News · Posted: Apr 14, 2018 10:00 AM MT | Last Updated: April 14, 2018
From the daikon-topped Dandy hot dog to spaghetti alla vodka and chicken liver toast, Dandy is pushing the boundaries of typical brewery fare. (Julie Van Rosendaal/CBC)
One of Calgary's first nano-breweries (even smaller than a microbrewery) opened its doors in August of 2014.
Since then, the Dandy Brewing Company has outgrown its tiny, industrial taproom on 25th Avenue N.E. and has opened up in Ramsay, taking over a space on 11th Street S.E. that was previously a motorcycle shop and garage.
The space is more than just a brewery and tasting room, even if it's still referred to by that name.
Dandy Brewing Company co-founder Ben Leon, with head brewer Dylan Nosal and chef Merritt Gordon, celebrate the opening of their new tasting room in Ramsay. (Julie Van Rosendaal/CBC)
"We still call it a tasting room," said managing director Ben Leon, one of four co-founders along with head brewer Dylan Nosal, Matt Gaetz and Derek Waghray.
"But it's classified as a restaurant, which mainly means we can serve other people's booze. We can bring in guest taps and serve cocktails and wine for those coming in with friends who might not like beer. It's a little more flexible with patio extensions and events too."
Brewery scene is booming
Dandy adds to a growing community of craft breweries in the area. Further north, Cold Garden has been a popular rec room-style hangout since opening a couple years ago, and High Line is tucked away in a parking lot on Ninth Avenue S.E. in Inglewood.
Coming soon, Ol' Beautiful Brewery plans to open in the old Smithbilt Hats building beside Cold Garden, Revival Brewcade is opening in the space that was previously the Gummi Boutique on Ninth Avenue, and '88 Brewing plans to open its doors this summer on Portland Street, across from the Crossroads Market.
The converted motorcycle garage is airy and open, featuring work from local artists, including painted cut-outs by Kelsey Fraser, who also designed a number of Dandy's beer labels. (Julie Van Rosendaal/CBC)
Historically, nearby Inglewood was known as the brewery district. The Calgary Brewing and Malting Company, now known as the old Molson site, was founded in 1893 and shut down in 1994.
Further south, the Manchester/Highfield industrial area is being rebranded as the Barley Belt due to their high density of craft breweries, including Banded Peak, the Annex Ale Project, Paddy's Barbecue & Brewery and Village Brewery just off Highfield Road.
Calgary's first craft cider company, the Uncommon Cider Co., is looking at a space in the area as well.
Calgary vegan YouTuber brings plant-based Korean spin-offs from screen to plate
"The city was great about developing the tasting room licence for breweries," said Leon. "They made it so much easier for breweries to have a tasting room — it automatically came with the brewery designation. But we were really set on having a restaurant here, to be able to provide more of an experience."
In the kitchen, Chef Merritt Gordon has cooked in kitchens of well-known restaurants across Canada, including Rouge, Park and Au Pied de Cochon in Montreal, and the Eden in the Rimrock Hotel (the only AAA/CAA Five-Diamond designated restaurant in western Canada) in Banff. He jumped at the opportunity to have his own kitchen and free rein with a brand new menu.
Dandy's unique brews are served in 5 oz or 12 oz pours to retain their quality to the bottom of the glass. (Julie Van Rosendaal/CBC)
Beyond the usual bar nibbles, which are anything but the usual — think chicken liver toast, pickled eggs, sweet potato ceviche and Welsh rarebit — they do a fantastic Dandy hot dog dressed with daikon slaw and crispy shallots. Platters of spaghetti alla vodka, roasted lamb shoulder, homemade pineapple molasses cake and warm maple pouding chômeur for dessert are also on the menu. (Pastry is one of Gordon's specialties.)
They plan to launch a brunch menu on the weekend of April 21.
One side of the L-shaped building houses their new tanks (they've kept their old brewery space as well), and the room is open and modern.
Work from local artists decks the walls, including painted cut-outs by Kelsey Fraser, who has also designed a number of their beer labels.
Here's the scoop on that new ice cream shop in Auburn Bay
Sunlight streams in from the repair shop's floor-to-ceiling garage doors, illuminating the movable tables, chairs and benches along one wall hand-built by Leon's father-in-law.
They serve their unique brews in five- or 12-ounce pours to retain their quality to the bottom of the glass.
"Our head brewer Dylan comes from fine dining as well, so he approaches beer from a different angle than a lot of brewers," said Leon.
"He comes at it from a flavour profile first."
'We still call it a tasting room. But it's classified as a restaurant, which mainly means we can serve other peoples' booze. We can bring in guest taps and serve cocktails and wine for those coming in with friends who might not like beer." (Julie Van Rosendaal/CBC)
The Uncontrollable Urge has soft coriander and sea salt, and the Pink Boots Brew #2 is a bold Belgian with a pink tinge from being brewed with hibiscus.
This summer, they hope to find a way to start a bike-powered growler delivery service for people in the community, reminiscent of milkman-style deliveries or vintage ice cream trucks.
"Everything is meant to be eaten with every beer — it's fine dining drinking food," said Leon. "That's the idea of the space in general. We're a beer company, but it's really about having a great overall experience."
MORE FOOD AND THE CITY | Chai Pani brings Bombay street food to Bridgeland
MORE FOOD AND THE CITY | Cilantro gets a 30th birthday makeover
Julie Van Rosendaal
Calgary Eyeopener's food guide
Julie Van Rosendaal talks about food trends, recipes and cooking tips on the Calgary Eyeopener every Tuesday at 8:20 a.m. MT. The best-selling cookbook author is a contributing food editor for the Globe and Mail, and writes for other publications across Canada.
Bookmark cbc.ca/juliesrecipes
Read her blog, Dinner with Julie
Follow Julie on Instagram
Follow Julie on Twitter


text:
Kryon Systems Announces Innovative Cloud-Based Robotic Process Automation Platform as a Service
New PaaS Offering by Virtual Workforce Leader Requires No Backend Integration with Supported Applications or IT Infrastructure Reconfiguration.
NEW YORK, February 22th, 2018 — Today, Kryon Systems, the leading provider of intelligent Robotic Process Automation (RPA) solutions, has announced an RPA Platform as a Service (PaaS) available on the cloud. This new level of accessibility enables Kryon's customers to plug into intelligent Robotic Process Automation activities through an enterprise-ready platform. Kryon's cloud-based offering allows enterprises to access and expand workplace automation processes via web-based internet browsers and applications. Businesses will now be able to implement Kryon's technology immediately through the cloud without technical infrastructure configuration traditional to the RPA space, saving time and costs in implementation.
Kryon's RPA cloud solution will help the enterprise to decrease the overhead costs and efforts related to infrastructure deployments, maintenance, configuration and support. Cloud accessibility will allow the robots to be more easily deployed, while better serving virtual environments with no footprint on an organization's core business infrastructure. The platform is flexible and extendable, can be installed on premises, or on a private or public cloud. Kryon's out-of-the-box ready PaaS is supported through online or onsite training.
"To be a partner for global enterprises, we at Kryon know that RPA technology must not only move at the speed of business, but also exceed it," said CEO of Kryon Systems, Harel Tayeb. "RPA is already showing significant results in decreasing operational costs and the benefits of shifting towards cloud services will provide even more value and increased flexibility to grow and scale without overhauling existing IT infrastructure."
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text:
Boissise-la-Bertrand est une commune française située dans le département de Seine-et-Marne, en région Île-de-France.

Géographie

Localisation
Boissise-la-Bertrand est située à  à l'ouest de Melun.

Communes limitrophes

Géologie et relief
La commune est classée en zone de sismicité 1, correspondant à une sismicité très faible. L'altitude varie de  à  pour le point le plus haut, le centre du bourg se situant à environ  d'altitude (mairie).
La majorité du territoire est composé de forêts, de bois et d'espaces verts.
La forêt de Bréviande occupe toute la partie nord, tandis qu'au sud la commune s'étend le long de la Seine.

Hydrographie

Le réseau hydrographique de la commune se compose de deux cours d'eau référencés :
 la Seine, fleuve long de , en limite sud-ouest de la commune ;
 le fossé 01 de la Commune du Mée-sur-Seine, , qui conflue avec la Seine.
La longueur totale des cours d'eau sur la commune est de .

Climat

La commune bénéficie d'un « climat océanique dégradé des plaines du Centre et du Nord », selon la typologie des climats en France  définie en 2010. Ce type affecte l'ensemble du Bassin parisien avec une extension vers le sud, dont la totalité des communes de Seine-et-Marne.

Les paramètres climatiques qui ont permis d'établir cette typologie comportent six variables pour les températures et huit pour les précipitations, dont les valeurs correspondent aux données mensuelles sur la normale 1971-2000. Les sept principales variables caractérisant la commune sont présentées dans l'encadré ci-contre. Avec le changement climatique, ces variables ont pu depuis évoluer. Cette évolution peut être constatée sur la station météorologique historique de Météo-France la plus proche, Melun - Villaroche, qui se trouve à  à vol d'oiseau, où la température moyenne annuelle évolue de  pour 1981-2010 à  pour 1991-2020.

Milieux naturels et biodiversité

Espaces protégés
La protection réglementaire est le mode d'intervention le plus fort pour préserver des espaces naturels remarquables et leur biodiversité associée.

Un  espace protégé est présent sur la commune :
la réserve naturelle régionale des « Bruyères de Sainte-Assise », d'une superficie de , classée en 2009, principalement du fait de ses intérêts floristiques et entomologiques.

Zones naturelles d'intérêt écologique, faunistique et floristique
L'inventaire des zones naturelles d'intérêt écologique, faunistique et floristique (ZNIEFF) a pour objectif de réaliser une couverture des zones les plus intéressantes sur le plan écologique, essentiellement dans la perspective d'améliorer la connaissance du patrimoine naturel national et de fournir aux différents décideurs un outil d'aide à la prise en compte de l'environnement dans l'aménagement du territoire.

Le territoire communal de Boissise-la-Bertrand comprend une ZNIEFF de ,
les « Landes de Ste-Assise et bois de Boissise-la-Bertrand » (), couvrant  du département
, et un ZNIEFF de ,
les « Bois et landes entre Seine-Port et Melun » (), couvrant  du département.

Urbanisme

Typologie
Boissise-la-Bertrand est une commune urbaine, car elle fait partie des communes denses ou de densité intermédiaire, au sens de la grille communale de densité de l'Insee.
Elle appartient à l'unité urbaine de Paris, une agglomération inter-départementale regroupant  et  en 2017, dont elle est une commune de la banlieue.

Par ailleurs la commune fait partie de l'aire d'attraction de Paris dont elle est une commune de la couronne. Cette aire regroupe .

Lieux-dits et écarts
La commune compte  lieux-dits administratifs répertoriés consultables ici (source : le fichier Fantoir).

Occupation des sols
L'occupation des sols de la commune, telle qu'elle ressort de la base de données européenne d'occupation biophysique des sols Corine Land Cover (CLC), est marquée par l'importance des forêts et milieux semi-naturels (73,7 % en 2018), en augmentation par rapport à 1990 (66,9 %). La répartition détaillée en 2018 est la suivante :
forêts (64% ), zones urbanisées (10,5% ), milieux à végétation arbustive et/ou herbacée (9,7% ), zones industrielles ou commerciales et réseaux de communication (7,1% ), eaux continentales (4,7% ), terres arables (3,9 %).

Parallèlement, L'Institut Paris Région, agence d'urbanisme de la région Île-de-France, a mis en place un inventaire numérique de l'occupation du sol de l'Île-de-France, dénommé le MOS (Mode d'occupation du sol), actualisé régulièrement depuis sa première édition en 1982. Réalisé à partir de photos aériennes, le Mos distingue les espaces naturels, agricoles et forestiers mais aussi les espaces urbains (habitat, infrastructures, équipements, activités économiques, etc.) selon une classification pouvant aller jusqu'à 81 postes, différente de celle de Corine Land Cover. L'Institut met également à disposition des outils permettant de visualiser par photo aérienne l'évolution de l'occupation des sols de la commune entre 1949 et 2018.

Planification
La loi SRU du  a incité les communes à se regrouper au sein d'un établissement public, pour déterminer les partis d'aménagement de l'espace au sein d'un SCoT, un document d'orientation stratégique des politiques publiques à une grande échelle et à un horizon de  et s'imposant aux documents d'urbanisme locaux, les PLU (Plan local d'urbanisme). La commune est dans le territoire du SCOT Région melunaise, dont l'élaboration a été engagée de 2013 à 2015, puis poursuivie à partir de 2017 sur un périmètre différent et porté par la communauté d'agglomération Melun Val de Seine.

La commune, en 2019, avait engagé l'élaboration d'un plan local d'urbanisme.

Logement
En 2016, le nombre total de logements dans la commune était de  dont 92,6 % de maisons et 7,4 % d'appartements.

Parmi ces logements, 86,4 % étaient des résidences principales, 5,5 % des résidences secondaires et 8,2 % des logements vacants.

La part des ménages fiscaux propriétaires de leur résidence principale s'élevait t à 84,1 % contre 15,1 % de locataires dont, 0,3 % de logements HLM loués vides (logements sociaux) et, 0,8 % logés gratuitement.

Voies de communication et transports
La commune est desservie par le réseau de bus Melibus qui dessert principalement la communauté d'agglomération Melun Val de Seine dont elle fait partie. Les lignes J et Jd permettent aux habitants de la ville de rejoindre le centre-ville de Melun via la gare du Mée.

Toponymie
 Du latin « boscus », le bois, et « sedes », la résidence.

Le nom de Boissise-la-Bertrand viendrait soit de Bertrand du Guesclin, de passage avec Jeanne d'Arc pour mettre le siège devant Melun, soit des noms accolés des deux châteaux médiévaux du village

Histoire

La plus ancienne trace historique trouvée à Boissise est un menhir trouvé dans le bois des Joies, appelé le Grand Berger. Ce menhir se trouve en réalité sur le territoire de la commune de Vert-Saint-Denis, près de la ferme du Gros Chêne. Les premiers documents écrits quant à eux datent du . En 1135 des religieux viennent bâtir un oratoire dédié à saint Acire dans le bois de Beaulieu, tandis qu'en 1172 un jugement du parlement de Paris évoque les habitants d'un lieu appelé Busseis, ce qui signifie « boccus sedes ». Boissise appartient aux barons de Seine-Port, avant de passer aux mains de la famille Vaudetar, qui conservera la paroisse jusqu'en 1720 et sa vente à Jean-Baptiste Glucq baron de Saint-Port.

Pendant la période de la Terreur, le mathématicien, astronome et physicien français Pierre-Simon de Laplace, qui avait quitté Melun pour Le Mée fut arrêté par les gardes nationaux de Boissise-la-Bertrand mais relâché aussitôt.

Politique et administration

Rattachements administratifs et électoraux
La commune se trouve dans l'arrondissement de Melun du département de Seine-et-Marne. Pour l'élection des députés, elle fait partie de la première circonscription de Seine-et-Marne.

Elle a été, de 1793 à 1801, chef-lieu de canton. En 1801, elle intègre le canton de Melun-Nord, qui est scindé en 1975 et la commune est rattachée au canton de Savigny-le-Temple. En 1991, elle fait partie du nouveau canton du Mée-sur-Seine, créé du fait de la forte croissance démographique du secteur consécutive à l'implantation de la ville nouvelle de Sénart, puis, dans le cadre du redécoupage cantonal de 2014 en France, elle est désormais intégrée au canton de Savigny-le-Temple.

Intercommunalité
La commune est membre de la communauté d'agglomération Melun Val de Seine (CAMVS), qui a succédé au District Melun – Le-Mée-sur-Seine qu'elle avait rejoint en 1994.

Liste des maires

Équipements et services

Eau et assainissement
L'organisation de la distribution de l'eau potable, de la collecte et du traitement des eaux usées et pluviales relève des communes. La loi NOTRe de 2015 a accru le rôle des EPCI à fiscalité propre en leur transférant cette compétence. Ce transfert devait en principe être effectif au , mais la loi Ferrand-Fesneau du  a introduit la possibilité d'un report de ce transfert au .

Assainissement des eaux usées
En 2020, la gestion du service d'assainissement collectif de la commune de Boissise-la-Bertrand est assurée par la communauté d'agglomération Melun Val de Seine (CAMVS) pour la collecte, le transport et la dépollution. Ce service est géré en délégation par une entreprise privée, dont le contrat arrive à échéance le .

L'assainissement non collectif (ANC) désigne les installations individuelles de traitement des eaux domestiques qui ne sont pas desservies par un réseau public de collecte des eaux usées et qui doivent en conséquence traiter elles-mêmes leurs eaux usées avant de les rejeter dans le milieu naturel. La communauté d'agglomération Melun Val de Seine (CAMVS) assure pour le compte de la commune le service public d'assainissement non collectif (SPANC), qui a pour mission de vérifier la bonne exécution des travaux de réalisation et de réhabilitation, ainsi que le bon fonctionnement et l'entretien des installations.

Eau potable
En 2020, l'alimentation en eau potable est assurée par la commune qui en a délégué la gestion à l'entreprise Veolia, dont le contrat expire le .

Population et société

Démographie

À la Révolution (1789), le village compte 303 habitants, et conserve une population stable tout au long du siècle suivant, avec 314 habitants en 1870. Le village est touché par les pertes humaines de la Première Guerre mondiale, et sa population décroît légèrement jusqu'à la Seconde Guerre mondiale. La courbe s'inverse à partir de 1945, une population de citadins remplaçant progressivement les anciens cultivateurs et vignerons. La ville passe à 418 habitants en 1968, puis à 571 en 1975. La croissance de la population continue sur les trente dernières années, avec 719 habitants en 1981, 788 en 1989 et 895 au recensement de 1999. Le cap des 1000 habitants est passé en 2011, année où l'on recense  habitants.

Enseignement
Boissise-la-Bertrand dispose d'une école élémentaire "Les Fontaines", située rue de la Fonaine-la-Reine.
Cet établissement public, inscrit sous le code UAI (Unité administrative immatriculée) : 0772672L, comprend   (chiffre du Ministère de l'Éducation nationale).

Il ne dispose pas d'un restaurant scolaire.

La commune dépend de l'Académie de Créteil ; pour le calendrier des vacances scolaires, Boissise-la-Bertrand est en zone C.

Économie

Revenus de la population et fiscalité
En 2017, le nombre de ménages fiscaux de la commune était de , représentant  et la  médiane du revenu disponible par unité de consommation  de  euros.

Emploi
En 2017, le nombre total d'emplois dans la zone était de , occupant   résidants.

Le taux d'activité de la population (actifs ayant un emploi) âgée de 15 à 64 ans s'élevait à 70,7 % contre un taux de chômage de 4,8 %.

Les 24,6 % d'inactifs se répartissent de la façon suivante : 9,6 % d'étudiants et stagiaires non rémunérés, 10,7 % de retraités ou préretraités et 4,3 % pour les autres inactifs.

Entreprises et commerces
En 2015, le nombre d'établissements actifs était de  dont   dans l'industrie,  dans la construction,  dans le commerce-transports-services divers et   étaient relatifs au secteur administratif.

Ces établissements ont pourvu  postes salariés.

Secteurs d'activité
Au , Boissise-la-Bertrand compte  actifs : 0 dans l'agriculture, 1 dans l'industrie, 2 dans la construction, 7 dans le commerce et les services et 4 dans le secteur administratif.

Agriculture
Boissise-la-Bertrand est dans la petite région agricole dénommée le « Pays de Bière et Forêt de Fontainebleau », couvrant le Pays de Bière et la forêt de Fontainebleau. En 2010, aucune orientation technico-économique de l'agriculture ne se dégage sur la commune.

Si la productivité agricole de la Seine-et-Marne se situe dans le peloton de tête des départements français, le département enregistre un double phénomène de disparition des terres cultivables (près de  par an dans les années 1980, moins dans les années 2000) et de réduction d'environ 30 % du nombre d'agriculteurs dans les années 2010. Cette tendance n'est pas confirmée au niveau de la commune qui voit le nombre d'exploitations rester constant entre 1988 et 2010.
Le tableau ci-dessous présente les principales caractéristiques des exploitations agricoles de Boissise-la-Bertrand, observées sur une période de  :

Culture locale et patrimoine

Lieux et monuments
 L'église Saint-Germain-l'Auxerrois,  et  siècles.
 Au centre radio-électrique de Sainte-Assise, à la limite avec Seine-Port, se trouve un groupe de 10 pylônes atteignant  de haut, servant entre autres à la transmission d'informations à destination des sous-marins français. Une partie du site est maintenant devenu la réserve naturelle régionale des Bruyères de Saint-Assise.
 Le château de Boissise, dont l'histoire est connue depuis le début du , propriété alors du receveur général des fermes Alexandre-Jean-Roch Colin de Saint-Marc et de son épouse (petite-fille de Pierre-Nicolas Hébert), a été acquis en 1807 par Charles Lucas « comte » Pinckney Horry, compagnon d'armes de La Fayette et propriétaire de la . Ce dernier avait épousé de Marie Florimonde de Fäy de La Tour-Maubourg, fille du général-comte, Charles César de Fay de La Tour-Maubourg. Le château a été transmis tout au long du  dans la même fratrie : d'abord à Armand Charles de La Tour-Maubourg, puis à Marie Stéphanie, comtesse Andréossy, enfin à Rodolphe de Faÿ de La Tour-Maubourg, général de division et pair de France, décédé à Boissise le 27 mai 1871. Ce dernier a donné son nom à une partie de la rue principale du centre-bourg.
 Le château de Beaulieu, du , qui accueillit successivement l'éditeur Ernest Panckoucke et l'homme d'affaires Edmond Régnier.
 L'Ermitage de Beaulieu, pavillon construit au , ancienne demeure d'Edgar Faure et de son épouse Lucie, actuellement résidence pour personnes âgées.
 Le nouveau « barrage de Vives-Eaux », avec une passerelle piétonne accessible au public permettant l'accès à Boissise-le-Roi, a été mis en service à l'automne 2018.

Patrimoine naturel
 Réserve naturelle régionale des bruyères de Sainte-Assise partagée avec Seine-Port.

Personnalités liées à la commune
 Clément-Pierre Marillier (1740-1808), dessinateur et graveur, fut propriétaire du château du  du hameau de Beaulieu-sur-Seine et, lorsque le découpage administratif de la Convention nationale fit de Boissise-la-Bertrand le chef-lieu de canton, il en fut le premier juge de paix.
 Rodolphe de Faÿ de La Tour-Maubourg (1787-1871), général et homme politique.
 Edmond Régnier (1822-1886) possédait une propriété au hameau de Beaulieu à la fin du Second Empire.
 Dominique Aury, alias Pauline Réage (1907-1998), y a vécu à la fin de sa vie, au 7 rue François Rolin, dans une maison acquise à la fin des années cinquante grâce aux droits d'Histoire d'O. Elle y a également accueilli Jean Paulhan malade.
 Lucie Faure (1908-1977), femme de lettres française, romancière, directrice de revue y est décédée. Elle et son mari, Edgar Faure homme politique, y eurent une villégiature.

Voir aussi

Bibliographie

 Louis Michelin, Essais historiques statistiques sur le département de Seine-et-Marne, Paris, Dumoulin, 1841.

Sources archivistiques
Les Archives nationales conservent sous la cote  MC/ET/XXIX/1519, dossier 19, plusieurs centaines de pièces (adjudications, comptes, ventes, inventaires après décès, décharges, renonciations, dépôts de testaments, délivrances de legs, titres de propriété) relatives à histoire du domaine de Boissise-la-Bertrand. Ces documents, dont le plus ancien remonte à 1692, sont librement consultables.

Articles connexes

 Liste des communes de Seine-et-Marne
 Communauté d'agglomération Melun-Val de Seine

Liens externes

 Boissise-la-Bertrand sur le site de la CA Melun-Val de Seine

Notes et références

Notes et cartes
 Notes

 Cartes

Références

Commune en Seine-et-Marne
Unité urbaine de Paris
Aire urbaine de Paris


text:
Survivor Nicaragua
Jane won't talk about "incident" that affected her vote; report says it was quid pro quo claim
by Andy Dehnart 19 Dec. 2010 | 7:45 pm
Perhaps the most interesting thing to happen on Survivor Nicaragua happened off-camera, but it had some impact and consequence, at least according to a report.
In an exit interview, Jane Bright told Fancast's Gordon Holmes that she voted for Sash "[b]ecause of an incident that happened that night prior to Tribal Council." Asked about it, Jane says, "I can't talk about it. But I was saying you guys are making a big mistake. I'm writing his name down even though I know he has an idol. Cause he is a big rattlesnake. And if y'all don't know it, you know it now. I'm not wasting my vote on Holly."
Survivor spoiler missyae uses his insider access to report that the incident involved a proposed deal:
"Jane accused Sash of offering to pay off her morgage in exchange for her vote and influence with the jury on his behalf. When she said this Jeff ordered the cameras to stop rolling and called in the producers. They discussed this for about 10 minutes and resumed filming without giving any instructions to the jury or the remaining players. In other words they left it as a 'he said, she said' situation."
Offering to pay another contestant's mortgage off would absolutely be against the rules, as a quid pro quo arrangement is explicitly prohibited by the Survivor rule book (see page 2, section A, number 5). That includes "a benefit or some form of consideration … after the Series has concluded." In other words, an agreement to share the prize money alone isn't the only thing that's prohibited.
Had that actually happened–again, it's he said/she said–the rules say that it "may, in Producer's sole discretion, result in that Contestant's expulsion from the Series Location and competition and/or that Contestant's forfeiture of the Prize and/or any other prizes, even if already awarded." The key word, of course, is "may."
RuPaul's Drag Race and Guy's Grocery Games: how their sets were designed
Reality TV production designer Scott Storey shares behind-the-scenes details about his designs of two of cable TV's most well-known reality competitions.
Big Brother behind the scenes
Answers to questions about what happens behind the scenes of CBS' Big Brother: from the casting process to live feed cuts, producer manipulation to what's behind the mirrors in the house.
Big Brother 12 house: inside the soundstage
Big Brother behind the scenes: a detailed tour of the soundstage home of the season 12 houseguests, including their bedrooms, bathroom, kitchen, and more.
Jeff Probst on a NaOnka return, Survivor Nicaragua ("not one of our greatest seasons")
Fabio's win sort of redeems Survivor Nicaragua; quitters may no longer be on jury
more stories about Survivor Nicaragua


text:
\section{Introduction}
We introduce the isoGeometric Residual Minimization (iGRM) method, which results in a stable \revMP{discretization} for  advection-dominated diffusion problems. The method applies the residual minimization technique to stabilize the discrete solution.
The minimum residual methods aim to find $u_h\in U_h$ such that
$$
u_h=\arg\!\!\!\min_{w_h\in U_h}\|b(w_h,\cdot) - \ell(\cdot)\|_{V^*}\,\,,
$$
where $U$ and $V$ are Hilbert spaces, $V^*$ denotes the dual of $V$, $b:U\times V\to \mathbb R$ is a continuous bilinear (weak) form, $U_h\subset U$ is a discrete solution space, and $\ell\in V^{*}$ is a given right-hand side. Several discretization techniques are particular incarnations of this wide-class of residual minimization methods. These include: the \emph{least-squares finite element method}~\cite{ LSFEM}, the \emph{discontinuous Petrov-Galerkin method (DPG) with optimal test functions}~\cite{ t8}, the \emph{variational stabilization method}~\cite{ VSM}, adaptive stabilized finite element method~\cite{ Sergio}, or the \emph{automatic variationally stable finite element method}~\cite{ romkes}.
\revMP{This residual minimization technique exploits the tensor product structure of the discrete space to deliver Uzawa-like iteration schemes to solve the resulting global system. We approach the residual minimization as a saddle point (mixed) formulation, as described in~\cite{Erikkson}. }

The residual minimization method relies on the selection of the residual space and its inner product. In particular, we select a weighted $H^1$ norm as the inner product to obtain a Kronecker product structure of the Gramm matrix.
\revMP{This structure enables us to obtain a linear computational cost preconditioner for the Uzawa-like iterative solver. The resulting method converges well in the weighted $H^1$ norm.}

For spatial discretization, we use IsoGeometric Analysis (IGA)~\cite{ IGA}. IGA  uses B-splines or NURBS~\cite{ NURBS} basis functions to construct a Galerkin finite element method (FEM).
In particular, we use a tensor-product B-spline space to approximate the residual minimization method.

This structure enables for fast solution of the projection problem with isogeometric analysis~\cite{ CG1, CG2, Gao2014}.  A similar idea is applied to fast direction splitting solvers of explicit dynamics~\cite{ ED1, ED2, ED3, ED4, ED5}. Splitting methods deliver fast factorizations as discussed, among many other sources, in~\cite{ Samarskii, Quarteroni, ADS1, ADS2, ADS3, ADS4}. A modern version of this method solves different classes of problems~\cite{ Minev1, Minev2}.

\revMP{We exploit the Kronecker product structure of the approximated Gramm matrix to approximate the Schur complement, with iterative corrections in the outer loop. We approximate the inverse of the Schur complement system with a conjugate gradient solver (CG)~\cite{ nocedal2006conjugate} in the inner loop.}

\revcc{In this paper, we blend isogeometric analysis, residual minimization, and the alternating directions splitting solver.  We solve the isogeometric Residual Minimization (iGRM) with a direction splitting preconditioner. We solve stationary advection-diffusion problems.}
We analyze four stationary computational problems, including a problem with the manufactured solution, the Eriksson-Johnson model problem, and a vortical wind problem. We use the Streamline Upwind Petrov-Galerkin (SUPG) and Discontinuous Petrov-Galerkin methods as references to compare the performance of our method.

\revcc{The structure of the paper is the following. We start from the introduction of the iGRM method in Section 2. Next, in Section 3, we describe the general idea and the details of the iterative solver. In Section 4, we present the numerical results for the manufactured solution problem, Erikkson-Johnson model problem, and the vortical wind problem. We conclude with Section 5.}

\section{The Isogeometric Residual Minimization Method (iGRM)}

To simplify the discussion, we focus on a two-dimensional model problem in space. Nevertheless, the extension of the formulation to three-dimensions is straightforward.
\revcc{We introduce $b:U\times V\to \mathbb R$ a continuous bilinear (weak) form defined over solution $U$ and residual $V$ Hilbert spaces, and $\ell\in V^{*}$ a given right-hand side, where $V^*$ denotes the dual of $V$.}

\subsection{Residual minimization method for the global problem}

For a general weak problem: Find $u\in U$ such as
\begin{equation}
b(u,v)=l(v) \quad \forall v\in V
\label{eq:gen_weak}
\end{equation}
we define the operator
\begin{equation}
 B: U \rightarrow V^{*}
\end{equation}
such that
\begin{equation}
 \langle Bu , v \rangle_{V^{*} \times V} = b(u,v)
\end{equation}
to reformulate the problem as
\begin{equation}
 Bu - l = 0
\end{equation}
We minimize the residual
\begin{equation}
 u_h = \textrm{argmin}_{w_h \in U_h} {1 \over 2} \| Bw_h - l \|_{V^{*}}^2
\end{equation}
We define the Riesz representer as
\begin{equation}
 R_V \colon V \ni v \rightarrow (v,.) \in V^{*}
\end{equation}
We then project the problem back to $V$
\begin{equation}
 u_h = \textrm{argmin}_{w_h \in U_h} {1 \over 2} \| R_V^{-1} (Bw_h - l) \|_V^2
\end{equation}
\revcc{We use here the definition of the dual norm
\begin{equation}
\| f\|_{V^{*}}:=\textrm{sup} \{ \langle f,v\rangle_{V^{*}\times V}: v\in V, \|v\|_V=1 \}.
\end{equation}}
\revcc{The definition of the dual norm implicitly depends on the $V$ norm. Thus, a vital ingredient of a residual minimization method is the selection of a norm on $V$. The quality of the results strongly depends on the choice for the norm on $V$.} The minimum is attained at $u_h$ when the G\^ateaux derivative is equal to $0$ in all directions:
we define the operator
\begin{equation}
\left( R_V^{-1} (Bu_h - l), R_V^{-1}(B\, w_h) \right)_V = 0 \quad \forall \, w_h \in U_h
\end{equation}
\revcc{We define the Riesz representation of the residual $r=R_V^{-1}(Bu_h-l)$. Thus, our problem reduces to
\begin{equation}
 \left( r , R_V^{-1} (B\,  w_h )  \right)_V = 0 \quad \forall \, w_h \in U_h
\end{equation}
The action of the inverse of the Riesz operator on the functional $Bw_h$ is equal to that element itself in $V$. Therefore, we get
\begin{equation}
 \langle Bw_h, r  \rangle_{V^{*}\times V} = 0 \quad \quad \forall w_h \in U_h.
\end{equation}}
From the definition of the error representation in terms of the residual, we have
\begin{equation}
(r,v)_V=\langle B u_h-l,v \rangle_{V^{*}\times V} \quad \forall v\in V.
\end{equation}
Thus, our problem reduces to the following semi-infinite problem: Find $(r,u_h)\in{V\times U_h}$ such that
\begin{eqnarray}
\begin{aligned}
(r,v)_V - \langle B u_h-l,v \rangle_{V^{*}\times V} &= 0 \quad  \forall v\in V \\
\langle Bw_h,r\rangle_{V^{*}\times V} &= 0 \quad  \forall w_h \in U_h
\end{aligned}
\end{eqnarray}
We discretize the residual space as $V_h \in V$ to \revMP{obtain} the discrete problem: Find $(r_h,u_h)\in{V_h\times U_h}$ such as
\begin{eqnarray}
\begin{aligned}
(r_h,v_h)_{V} - \langle B u_h-l,v_h \rangle_{V^{*}\times V} &= 0 \quad  \forall v_h\in V_h \\
\langle Bw_h,r_h\rangle_{V^{*}\times V} &= 0 \quad \forall w_h \in U_h
\end{aligned}
\label{eq:resmin}
\end{eqnarray}
where $(\cdot,\cdot)_{V}$ is the inner product in $V_h$ inherited from $V$, $\langle Bu_h,v_h\rangle_{V^{*}\times V} = b\left(u_h,v_h\right)$, $\langle Bw_h,r_h\rangle_{V^{*}\times V} = b\left(w_h,r_h\right)$.
\begin{remark}
We define the discrete residual space $V_h$ to be sufficiently close to the continuous $V$ space to ensure stability. Indeed, we gain stability by enriching the residual space $V_h$ (while fixing the solution space $U_h$) until we reach the discrete inf-sup stability.
\end{remark}

\subsection{Minimal residual discretization for the global problem with
B-splines}

We approximate the solution with a tensor product of one-dimensional B-splines. We choose a uniform order $p$ in all directions to simplify the discussion.

We denote the basis functions in the $x$-direction for the discrete solution and residual spaces as $n_a$ and $N_A$, respectively.  Similarly, we denote the basis functions in the $y$-direction for the discrete solution and residual spaces as $m_b$ and $M_B$, respectively.

\revMP{We} approximate the residual space with a \revMP{sufficiently} larger discrete space \revMP{than} the solution space to guarantee discrete stability (cf.~\cite{Sergio}). We achieve this by either increasing the polynomial order or decreasing the continuity of the residual space to define the solution space.  As a result, we now can define the discrete solution $u_h$ and residual $w_h$ functions as:
\begin{equation}
\begin{aligned}
  u_h &= \sum_{a,b} w_{ab} n_{a}m_{b} \\
  w_h &= \sum_{A,B} u_{AB} N_{A}M_{B}

\end{aligned}
\end{equation}
and the discrete solution $r_h$ and residual $v_h$ functions as
\begin{equation}
\begin{aligned}
r_h &= \sum_{a,b} r_{ab} n_{a}m_{b} \\
v_h &= \sum_{A,B} v_{AB} N_{A}M_{B}

\end{aligned}
\end{equation}

\newcommand{\partial K}{\partial K}
\newcommand{\F_{\dK}}{\mathcal{F}_{\partial K}}
\newcommand{Z_{\F,0}}{Z_{\mathcal{F},0}}
\newcommand{Z_{\F,0}^\dagger}{Z_{\mathcal{F},0}^\dagger}
\newcommand{V_\K^{k,\mathrm{c}}}{V_\mathcal{K}^{k,\mathrm{c}}}
\newcommand{e_\K^{k,\mathrm{c}}}{e_\mathcal{K}^{k,\mathrm{c}}}

\section{Conjugate Gradients method for Schur complement}

In this section, we derive an iterative solution algorithm for the residual minimization problem. We denote by $\Omega$ a bounded open set of $\mb{R}^d$ with Lipschitz continuous boundary $\partial \Omega$, where $d=2,3$.  \revMP{We} assume that the domain $\Omega = \Omega_x \times \Omega_y$ has a tensor product structure. This structure allows us to approximate the Gramm matrix as a Kronecker product of one-dimensional matrices.

The resulting matrix system for the residual minimization method is
\begin{equation} \label{eq:mp}
\begin{bmatrix}
G & B \\
B^T & 0 \\
\end{bmatrix}
\begin{bmatrix}
r \\
u \\
\end{bmatrix}
=
\begin{bmatrix}
F \\
0 \\
\end{bmatrix},
\end{equation}
where we minimize the isogeometric discretization residual of~\eqref{eq:resmin} in the energy norm $G$ (a weighted $H^1$ norm), where $G$ is defined as
\begin{equation} \label{eq:A}
G = M + \eta K
\end{equation}
\revMP{where we keep the weight $\eta$ as a solver parameter, and }
\revcc{\begin{equation}
\begin{aligned}
M & = \int_{\Omega_x} \int_{\Omega_y}  n_a m_b N_C M_D \,dx \,dy \\
& = \int_{\Omega_x} n_a N_C \,dx \int_{\Omega_y} m_b M_D \,dy = M_x \otimes M_y, \\
K & = K_x \otimes M_y + M_x \otimes K_y,
\end{aligned}
\end{equation}
where $M_x, M_y$, and $K_x, K_y$ denote 1D mass matrices and stiffness matrices, respectively, defined as
\begin{equation}
\begin{aligned}
\{ M_x \}_{aC} &= m\left( n_a , N_C \right)= \int_{\Omega_x} n_a N_C \, dx \\
\{ M_y \}_{bD} &= m\left( n_b , N_D \right)=\int_{\Omega_y} m_b M_D \, dy \\
\{ K_x \}_{aC} &= a\left( n_a , N_C \right)=\int_{\Omega_x} \frac{d n_a}{dx} \frac{d N_C}{dx} \, dx \\
\{ K_y \}_{bD} &= a\left( m_b , M_D \right)=\int_{\Omega_y} \frac{d m_b}{dy} \frac{d M_D}{dy} \,dy. \\
\end{aligned}
\end{equation}}

\revcc{We approximate the Gramm matrix in the residual minimization as a Kronecker product matrix $\tilde{G} =\left(K_x+\eta M_x\right) \otimes \left(K_y + \eta M_y\right)$, that is inexpensive to solve \revMP{(it requires solutions of two one-dimensional problems with multiple right-hand sides, where the sizes of the problems and the number of right-hand-sides corresponds to the mesh dimensions in both directions)} and only introduces an error of order $\revc{\eta^2}$. From~\eqref{eq:A}, we have}
\begin{equation}
\begin{aligned}
G & =  (M_x +  \eta K_x) \otimes (M_y +  \eta K_y) - \eta^2 K_x \otimes K_y \nonumber\\
&=\tilde{G} - \tilde K.
\end{aligned} \label{eq:Atilde}
\end{equation}
where $\tilde K = \eta^2 K_x \otimes K_y$.

\revcc{The definition of the $B$ matrix in~\eqref{eq:mp} corresponds to applying an appropriate isogeometric weak form of the scalar field PDE.}

\subsection{Overview of the iterative algoritm}

\revcc{In this section, we describe the iterative algorithm. Given an initial guess $\begin{bmatrix}
r^k  \\
u^k \end{bmatrix}$, we compute the update as
\begin{equation} \label{eq:update}
\begin{bmatrix}
d  \\
c \\
\end{bmatrix} =
\begin{bmatrix}
r-r^k  \\
u-u^k \\
\end{bmatrix} \nonumber
\end{equation}}
\revcc{Thus
\begin{equation} \label{eq:get_update}
\begin{bmatrix}
G & B \\
B^T & 0 \\
\end{bmatrix}
\begin{bmatrix}
d \\
c \\
\end{bmatrix}
=
\begin{bmatrix}
F - G r^k - Bu^k \\
-B^T r^k \\
\end{bmatrix}  \nonumber
\end{equation}
This system of linear equations is expensive to factorize, so we use the approximate $\tilde{G}$
\begin{equation} \label{eq:solve_update}
\begin{bmatrix}
\tilde{G} & B \\
B^T & 0 \\
\end{bmatrix}
\begin{bmatrix}
d \\
c \\
\end{bmatrix}
=
\begin{bmatrix}
F - G r^k - Bu^k \\
-B^T r^k \\
\end{bmatrix} \nonumber
\end{equation}}
\revcc{We propose the following iterative algorithm}
\begin{algorithm} [H]
\caption{\revcc{Iterative algorithm}} \label{alg:alg1}
\begin{algorithmic}
  \State \revcc{Initialize $\{u^{0} = 0; r^{0} = 0 \}$} \\
  \revcc{\textbf{for }{$k=1 \ldots N$ until convergence}}
  \begin{equation}
    \begin{aligned}
      \revcc{\textrm{Compute Schur }} & \revcc{\textrm {complement with linear } {\cal O}(N)
      \textrm{ cost}}\\
      %
    \revcc{  \begin{bmatrix}
 & \tilde{G} & B \\
 & B^T       & 0 \\
      \end{bmatrix}
      \begin{bmatrix}
 & d^k \\ c^k
      \end{bmatrix}}
      & \revcc{=
      \begin{bmatrix}
 & F - G r^k - B u^k \\
 & -B^T r^k \\
      \end{bmatrix}} \\
      %
      & \revcc{\textrm{Solve}} \\
 \revcc{ B^T \tilde{G}^{-1} B\, c^k} & \revcc{=
  B^T \tilde{G}^{-1}(F + \tilde{K} r^k - B u^k)} \\
      &  \revcc{\textrm{with PCG or \revMP{Multi-frontal solver}}} \\
      \revcc{r^{k+1}} & \revcc{= d^k+r^k} \\
      \revcc{u^{k+1}} & \revcc{= c^k+u^k} \\
      \revcc{k} & \revcc{\gets k+1} \\
    \end{aligned}
    \label{eq:CGouter}
  \end{equation}
\end{algorithmic}
\end{algorithm}

\subsection{Convergence of the iterative algorithm}

Since the Algorithm~\ref{alg:alg1} uses a preconditioned CG and both the CG and preconditioned CG are convergent (see, for example,~\cite{ nocedal2006conjugate}), we focus on the spectral analysis of~\eqref{eq:CGouter} in Algorithm~\ref{alg:alg1}.

Applying the initialization $u^{(0)} = u^k$\revMP{,} we obtain
\begin{equation} \label{eq:cv1}
u^{k+1} = u^k + c^k
\end{equation}
where $c^k$ is the update of $u^k$.

We now have
\begin{equation} \label{eq:cv2}
r^{k+1} = \tilde{G}^{-1} ( F + \tilde K r^k  - B u^k) - \tilde{G}^{-1} B c^k.
\end{equation}

To simplify the spectral analysis and without loss of generality\revMP{,} we set $F=0$. Thus, combining~\eqref{eq:cv1} and~\eqref{eq:cv2} gives
\begin{equation} \label{eq:cvp}
\begin{bmatrix}
u^{k+1} \\
r^{k+1} \\
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 \\
-\tilde{G}^{-1}B & \tilde{G}^{-1} \tilde K \\
\end{bmatrix}
\begin{bmatrix}
u^k \\
r^k \\
\end{bmatrix}
+
\begin{bmatrix}
c^k \\
-\tilde{G}^{-1}B c^k \\
\end{bmatrix}.
\end{equation}

Now, we analyze the spectrum of
\begin{equation} \label{eq:A0}
\tilde{G}^{-1} \tilde K = (M_x +  \revc{\eta} K_x)^{-1} \otimes (M_y +  \revc{\eta} K_y)^{-1} \cdot \left(\revc{\eta} K_x \otimes \revc{\eta} K_y\right).
\end{equation}

We apply a spectral decomposition~\cite{ horn1990matrix} to the matrix $K_\xi, \xi=x,y$ with respect to $M_\xi$ and arrive at
\begin{equation} \label{eq:sd0}
K_\xi = M_\xi P_\xi D_\xi P_\xi^{-1},
\end{equation}
where $D_\xi$ is a diagonal matrix with entries to be the eigenvalues of the generalized eigenvalue problem
\begin{equation} \label{eq:eigKM0}
K_\xi v_\xi = \lambda_\xi M_\xi v_\xi
\end{equation}
and the columns of $P_\xi$ are the eigenvectors. We assume that all the eigenvalues are sorted in ascending order and are listed in $D_\xi$ and the $j$-th column of $P_\xi$ is associated with the eigenvalue $\lambda_{\xi,j} = D_{\xi,jj}$.

Using~\eqref{eq:sd0} and~\eqref{eq:A0}, we now calculate
\begin{equation}
\begin{aligned}
\tilde{G}^{-1} & = (M_x +  \revc{\eta} K_x)^{-1} \otimes (M_y +  \revc{\eta} K_y)^{-1} \\
& = (M_x + \revc{\eta} M_x P_x D_x P_x^{-1} )^{-1} \otimes (M_y + \revc{\eta} M_y P_y D_y P_y^{-1} )^{-1} \\
& = P_x E_x P_x^{-1} M_x^{-1} \otimes P_y E_y P_y^{-1} M_y^{-1},
\end{aligned}
\end{equation}
where
\begin{equation}
E_\xi = ( I + \revc{\eta} D_\xi )^{-1}, \qquad \xi = x, y.
\end{equation}
We assume here that the mesh is uniform in both directions.

Thus, similarly, we have
\begin{equation}
\begin{aligned}
\tilde{G}^{-1} \tilde K & = \big( P_x E_x  P_x^{-1} M_x^{-1} \otimes P_y E_y P_y^{-1} M_y^{-1} \big)  \\
& \qquad  \cdot \big( \revc{\eta} M_x P_x D_x P_x^{-1} \otimes \revc{\eta} M_y P_y D_y P_y^{-1} \big) \\
& = \big( P_x \otimes P_y \big) \cdot \big( \revc{\eta}E_x D_x \otimes \revc{\eta} E_yD_y  \big) \cdot \big( P_x^{-1} \otimes P_y^{-1} \big) \\
& = \big( P_x \otimes P_y \big) \cdot \big( \revc{\eta}( I + \revc{\eta} D_x )^{-1} D_x \otimes \revc{\eta} ( I + \revc{\eta} D_y )^{-1} D_y  \big) \cdot \big( P_x^{-1} \otimes P_y^{-1} \big). \\
\end{aligned}
\end{equation}

The middle term is a diagonal matrix. Thus, a typical eigenvalue of $\tilde A^{-1} \tilde K$ is
\begin{equation}
\lambda = \frac{\revc{\eta^2} \lambda_{x,i} \lambda_{y,j}}{ (1+\revc{\eta} \lambda_{x,i}) (1+\revc{\eta} \lambda_{y,i})},
\end{equation}
where $i,j$ are indices of eigenvalues in each dimension. The spectral radius of $\tilde{G}^{-1} \tilde K$ is then
\begin{equation}
\rho = \frac{\revc{\eta^2} \lambda_{x,\max} \lambda_{y,\max}}{ (1+\revc{\eta} \lambda_{x,\max}) (1+\revc{\eta} \lambda_{y,\max})},
\end{equation}
where $\lambda_{\xi,\max}, \xi = x,y$ are the maximum eigenvalues in each dimension.  Immediately, we have
\begin{equation}
0 < \lambda \le \rho <1.
\end{equation}

Thus, the eigenvalues of the amplifying block-matrix (the vector in terms of $c^k$ is from inner CG) in~\eqref{eq:cvp} are 1 and $\lambda$. Thus, they are \revMP{bounded} by 1, which implies that the eigenvalues of their powers are also bounded by 1. Hence, the iterative Algorithm~\ref{alg:alg1} is convergent.

\section{Numerical results for stationary problems}

\revcc{In this section, we focus on a model advection-diffusion problem defined as follows. For a unitary square domain $\Omega=(0,1)^2$, the advection vector $\beta$ and the diffusion coefficient $\epsilon$, we seek the solution of the advection-diffusion equation
\begin{equation}
\beta_x\frac{\partial u}{\partial x}+\beta_y\frac{\partial u}{\partial y}-\epsilon \left(\frac{\partial^2 u}{\partial x^2}+
\frac{\partial^2 u}{\partial y^2}\right)=f
\label{eq:ModelProblem}
\end{equation}
with Dirichlet boundary conditions $u=g$ on the whole boundary of $\Gamma=\partial\Omega$.  We partition the boundary $\Gamma=\partial \Omega$ into the inflow $\Gamma^{-}=\{ x\in\Gamma: \beta \cdot n<0 \}=\{(x,y): xy=0\}$ and the outflow $\Gamma^{+}=\{x\in\Gamma: \beta \cdot n \geq 0\}$.}

\revcc{We introduce the discrete weak formulation
\begin{equation}
b(u_h,v_h)=l(v_h) \quad \forall v\in V
\label{eq:ModelProblem_weak}
\end{equation}
\begin{eqnarray}
\begin{aligned}
b(u_h,v_h) &=
\beta_x\left(\frac{\partial u_h}{\partial x},v_h\right)_{\Omega}+\beta_y\left(\frac{\partial u_h}{\partial y},v_h\right)_{\Omega}+\epsilon \left(  \frac{\partial u_h}{\partial x}, \frac{\partial v_h}{\partial x}\right)_{\Omega} +\epsilon \left(  \frac{\partial u_h }{\partial y}, \frac{\partial v_h}{\partial y}\right)_{\Omega} \\
&\inbrown{-\left(\epsilon\frac{\partial u_h}{\partial x}n_x,v_h\right)_{\Gamma}
-\left(\epsilon\frac{\partial u_h}{\partial y}n_y,v_h\right)_{\Gamma}}  \nonumber\\
&\inred{-\left(u_h,\epsilon \nabla v_h \cdot n\right)_{\Gamma} +\left(u_h,\beta\cdot n v_h\right)_{\Gamma^{-}}}\ingray{-\sum_K\left(u_h,3p^2 \epsilon/h_K v_h\right)_{\Gamma|_K} } \nonumber
\label{eq:ModelProblem_b}
\end{aligned}
\end{eqnarray}
where $n=(n_x,n_y)$ is the \revMP{vector} normal to $\Gamma$, and $h_K$ is the \revMP{``normal''} elemental distance (i.e., mesh size in the \revMP{direction} of the normal) (possibly changing with elements $K$ of the mesh),
\begin{equation}
l(v_h)=(f,v_h)_{\Omega}\inred{-\left(g,\epsilon\nabla v_h \cdot n\right)_{\Gamma}
+\left(g,\beta\cdot n v_h\right)_{\Gamma^{-}}}-\sum_K\ingray{\left(g,3p^2 \epsilon/h_K v_h\right)_{\Gamma}}\revMP{,}
\label{eq:ModelProblem_l}
\end{equation}
where the \ingray{gray} and \inred{red} represent the penalty terms, and $f$ corresponds to the forcing term, while the \inbrown{brown} terms result from the integration by parts~\cite{ weak}.}

In our model problem, we seek the solution in space $U = V = H^1\left(\Omega\right)$. The inner product in $V$ is defined as weighted $H^1$ norm (\ref{eq:A}).

\revcc{A popular stabilization technique is the Streamline-upwind Petrov-Galerkin (SUPG) method~\cite{ SUPG2, SUPG}. This method modifies the weak form as follows
\begin{equation}
b(u_h,v_h) + \sum_K \inred{(R(u_h),\tau \beta\cdot \nabla v_h)_K}=l(v_h)
\revMP{+\sum_K \inblue{(f,\tau \beta\cdot \nabla v_h)_K}}
 \quad \forall v\in V
\label{eq:Erikkson_weak_SUPG}
\end{equation}
where $R(u_h)=\beta \cdot \nabla u_h +\epsilon \Delta u_h$, and $\tau^{-1}= \beta \cdot \left(\frac{1}{h^x_K},\frac{1}{h^y_K} \right) + 3p^2\epsilon \frac{1}{{h^x_K}^2+h{^y_K}^2}$, where $\epsilon$ stands for the diffusion term, and $\beta = (\beta_x,\beta_y)$ for the convection term, and $h^x_K$ and $h^y_K$ are horizontal and vertical dimensions of an element $K$.  Thus, we have
\begin{equation}
b_{\inred{SUPG}}(u_h,v_h)=l_{\inblue{SUPG}}(v_h) \quad \forall v_h\in V_h
\label{eq:SUPG_weak}
\end{equation}
\begin{eqnarray}
\begin{aligned}
b_{\inred{SUPG}}(u_h,v_h)=
& \left(\frac{\partial u_h}{\partial x},v\right)+\epsilon \left(  \frac{\partial u_h}{\partial x}, \frac{\partial v_h}{\partial x}\right) +\epsilon \left(  \frac{\partial u_h }{\partial y}, \frac{\partial v_h}{\partial y}\right)  \\
&-\left(\epsilon\frac{\partial u_h}{\partial x}n_x,v_h\right)_{\Gamma}
-\left(\epsilon\frac{\partial u_h}{\partial y}n_y,v_h\right)_{\Gamma}\nonumber\\
&-\left(u_h,\epsilon \nabla v_h \cdot n\right)_{\Gamma} -\left(u_h,\beta\cdot n v_h\right)_{\Gamma}-\left(u_h,3p^2 \epsilon/h v_h\right)_{\Gamma}  \nonumber
\\
&+\inred{\left(\frac{\partial u_h}{\partial x}+\epsilon \Delta u_h,
\left(\frac{1}{h_x}  + 3\epsilon \frac{p^2}{{h^x_K}^2+{h^y_K}^2}\right)^{-1} \frac{\partial v_h}{\partial x}
\right)}
\label{eq:Erikkson_b_SUPG}
\end{aligned}
\end{eqnarray}}
\revMP{\begin{eqnarray}
\begin{aligned}
l_{\inblue{SUPG}}(v_h)&=(f,v_h)-\left(g,\epsilon\nabla v_h \cdot n\right)_{\Gamma} \nonumber \\
&+\left(g,\beta\cdot n v_h\right)_{\Gamma^{-}}-\left(g,3p^2 \epsilon/h_K v_h\right)_{\Gamma}
\revMP{+\inblue{\left(f,\left(\frac{1}{h_x}  + 3\epsilon \frac{p^2}{{h^x_K}^2+{h^y_K}^2}\right)^{-1}\frac{\partial v_h}{\partial x}\right)}}
\revMP{.} \nonumber
\label{eq:ModelProblem_l}
\end{aligned}
\end{eqnarray}}

\subsection{A manufactured solution problem}

First, we select the advection vector $\beta=(1,1)^T$, and $Pe=1/\epsilon=100$ and solve the advection-diffusion equation (\ref{eq:ModelProblem}) with homogeneous Dirichlet boundary conditions.  We utilize a manufactured solution $u(x,y)=\left(x+\frac{e^{Pe*x}-1}{1-e^{Pe}}\right)\left(y+\frac{e^{Pe*y}-1}{1-e^{Pe}}\right)$ enforced by the forcing term $f$.  \revcc{This analytic expression of the solution limits the P\'eclet number to $Pe=100$ due to machine precision.}

We use the weak form (\ref{eq:ModelProblem_weak}) and the inner product (\ref{eq:A}) into the iGRM setup (\ref{eq:resmin}) and use the preconditioned CG solver described in Section 3.
\revcc{In the weighted $H^1$ \revMP{norm} we introduce $\eta=h^2$, where $h$ is the diameter of the element.}

In the following problem, we study the $h$- and $p$-convergence of the iGRM method on uniform grids using different combinations of solution and residual spaces. We do not employ adaptive Shishkin grids here~\cite{ Kopteva}.  We increase the accuracy by increasing the order and continuity of solution spaces ($k$-\revMP{increase}~\cite{IGA, IGA2}) simultaneously. \revMP{Increasing} solution space order and continuity improves the accuracy of the solution \revMP{for this smooth problem}. Similarly, refining the mesh also improves the accuracy of the solution.

Table~\ref{tab:Problem1} illustrates the $h$ and $p$-convergence of the method.  The rows represent $p$-refinement of the solution test, from $(p,p-1)$ (order $p$, continuity $p-1$) to $(p+1,p)$ (order $p+1$, continuity p), and the columns represent $h$-refinement, from $n\times n$ mesh to $2n \times 2n$ mesh.  We report the numbers of degrees of freedom and the error in $L^2$ and $H^1$ norms.

\revcc{The black colors on the numerical results represent unwanted negative values of the solution.  We can read from Table~\ref{tab:Problem1} that $k$-refinement (increasing polynomial order of the solution with maximum continuity) is more attractive than $h$-refinement since the solution improves using fewer degrees of freedom.}

\revcc{We also present in Table~\ref{tab:Problem1SUPG} the results for SUPG method for the corresponding meshes and approximation spaces. \revMP{We conclude that we can achieve less than 1 percent error}, as measured in $L^2$ and $H^1$ norms\revMP{,} for iGRM method for $\#DOF=5594$ for trial (5,4) test (2,0) over $32\times 32$ mesh, and for SUPG method for $\#DOF=4761$ for trial=test=(5,4) over $64\times 64 $ mesh.}

{\tiny\begin{table*}[htp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
n & trial (2,1)  & trial (3,2)  & trial (4,3) & trial (5,4) \\
 & test (2,0) & test (2,0) & test (2,0)  & test (2,0)  \\
\hline
\#DOF & 389 & 410 & 433 & 458 \\
L2 & 192.47 & 151.23 & 78.69 & 28.11 \\
H1 & 101.14 & 74.54 & 44.33 & 32.05 \\
$8\times 8$ &
\includegraphics[scale=0.16]{8_21_20.pdf}
&
\includegraphics[scale=0.16]{8_32_20.pdf}
&
\includegraphics[scale=0.16]{8_43_20.pdf}
&
\includegraphics[scale=0.16]{8_54_20.pdf}
\\
\hline
\#DOF & 1413 & 1450 & 1489 & 1530 \\
L2 & 80.01 & 16.64 & 3.29 & 1.48 \\
H1 & 59.56 & 29.83 & 18.04 & 10.40
\\
$16\times 16$ &
\includegraphics[scale=0.16]{16_21_20.pdf}
&
\includegraphics[scale=0.16]{16_32_20.pdf}
&
\includegraphics[scale=0.16]{16_43_20.pdf}
&
\includegraphics[scale=0.16]{16_54_20.pdf}
\\
\hline
\#DOF & 5381 & 5450 & 5521 & 5594 \\
L2 & 32.07 & 1.33 & 0.27 & 0.056 \\
H1 & 31.01 & 9.77 & 3.16 & 0.82
\\
$32\times 32$ &
\includegraphics[scale=0.16]{32_21_20.pdf}
&
\includegraphics[scale=0.16]{32_32_20.pdf}
&
\includegraphics[scale=0.16]{32_43_20.pdf}
&
\includegraphics[scale=0.16]{32_54_20.pdf}
\\
\hline
\#DOF & 20997& 21130& 21265 & 21402 \\
L2 & 7.66 & 0.07 & 0.01 & 0.003 \\
H1 & 9.86 & 1.67 & 0.26 & 0.068
\\
$64\times 64$ &
\includegraphics[scale=0.16]{64_21_20.pdf}
&
\includegraphics[scale=0.16]{64_32_20.pdf}
&
\includegraphics[scale=0.16]{64_43_20.pdf}
&
\includegraphics[scale=0.16]{64_54_20.pdf}
\\
\hline
\end{tabular}
\includegraphics[scale=0.4]{legend.pdf}
\end{center}
\caption{Solution of Problem 1 by iGRM method, with different solution and residual spaces, for different mesh dimensions.}
\label{tab:Problem1}
\end{table*}}

{\tiny\begin{table*}[htp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
n & trial (2,1)  & trial (3,2)  & trial (4,3) & trial (5,4) \\
 & test (2,1) & test (3,2) & test (4,3)  & test (5,4)  \\
\hline
\#DOF & 100 & 121 & 144 & 169 \\
L2 & 40.36 & 38.74 & 38.72 & 38.82 \\
H1 & 73.30 & 80.16 & 82.86 & 83.72 \\
$8\times 8$ &
\includegraphics[scale=0.15]{supg_8_21.pdf}
&
\includegraphics[scale=0.15]{supg_8_32.pdf}
&
\includegraphics[scale=0.15]{supg_8_43.pdf}
&
\includegraphics[scale=0.15]{supg_8_54.pdf}
\\
\hline
\#DOF & 324 & 361 & 400  & 441 \\
L2 & 20.04 & 19.00 & 18.61 & 18.52 \\
H1 & 63.04 & 62.19 & 61.24 & 61.01 \\
$16\times 16$ &
\includegraphics[scale=0.15]{supg_16_21.pdf}
&
\includegraphics[scale=0.15]{supg_16_32.pdf}
&
\includegraphics[scale=0.15]{supg_16_43.pdf}
&
\includegraphics[scale=0.15]{supg_16_54.pdf}
\\
\hline
\#DOF & 1156 & 1225 & 1296 & 1369 \\
L2 & 6.51 & 5.07 & 4.20 & 3.66 \\
H1 & 36.57 & 26.88 & 21.16 & 17.35 \\
$32\times 32$ &
\includegraphics[scale=0.15]{supg_32_21.pdf}
&
\includegraphics[scale=0.15]{supg_32_32.pdf}
&
\includegraphics[scale=0.15]{supg_32_43.pdf}
&
\includegraphics[scale=0.15]{supg_32_54.pdf}
\\
\hline
\#DOF & 4356 & 4489 & 4624 & 4761 \\
L2 & 1.19 & 0.80 & 0.75 & 0.75 \\
H1 & 11.50 & 3.94 & 1.39 & 0.62 \\
$64\times 64$ &
\includegraphics[scale=0.15]{supg_64_21.pdf}
&
\includegraphics[scale=0.15]{supg_64_32.pdf}
&
\includegraphics[scale=0.15]{supg_64_43.pdf}
&
\includegraphics[scale=0.15]{supg_64_54.pdf}
\\
\hline
\end{tabular}
\includegraphics[scale=0.4]{legend.pdf}
\end{center}
\caption{\revcc{Solution of Problem 1 by SUPG method, with different solution and residual spaces, for different mesh dimensions.}}
\label{tab:Problem1SUPG}
\end{table*}}

\subsection{Erikkson-Johnson model problem}

We focus now on the model Eriksson-Johnson problem with the modifications proposed by~\cite{ Erikkson}. For the square domain $\Omega=(0,1)^2$ and the advection vector $\beta=(1,0)^T$, we seek the solution of the advection-diffusion equation~\eqref{eq:ModelProblem}.  We introduce the Dirichlet boundary conditions
$$
u=g=sin(\pi y) \textrm{ for }x\in \Gamma^{-}
$$
$$
u=0 \textrm{ for } x\in \Gamma^{+}
$$
weakly on the boundary $\Gamma$.  The inflow Dirichlet boundary condition drives the problem and it develops a boundary layer of width $\epsilon$ at the outflow $x = 1$.

\revcc{In the manufactured solution problem, we compare the SUPG and iGRM methods on a sequence of uniform grids. \revMP{Later}, we compare iGRM and SUPG methods on a sequence of adapted grids.  We simulate the Erikkson-Johnson problem with the iGRM and SUPG methods with $Pe=10^6$ on a sequence of grids refined towards the boundary layer. We start from the uniform grid of $2 \times 2$ elements, and we add the knot points to the last interval on the right in the $x$ direction.  We refine the knot vector in the $x$ direction by breaking in half the rightmost element.  We continue to refine in the perpendicular direction after we capture the boundary layer when the element size in one direction is \revMP{below} $\epsilon$.}

\revcc{We illustrate the solutions obtained from the iGRM and SUPG methods in Figures~\ref{fig:Crosses1}-\ref{fig:Crosses3}. We report the convergence in \revMP{$L^2$} and \revMP{$H^1$} norms in Table~\ref{tab:iGRMSUPG_E}.}

We now compare \revMP{iGRM vs} the DPG method~\cite{ Erikkson} using Lagrange (2,0) polynomials for solution and (3,0) polynomials for the residual. In our method, we use a (2,1) space for the solution with either a (3,1) or (3,2) space for the residual. Thus, our solution spaces have identical order and higher continuity, so they are smaller than those used in~\cite{ Erikkson}.

\revcc{We compare the iGRM computations with the DPG results from Figure 5.3 in~\cite{ Erikkson}, which shows the $L^2$ errors. Both iGRM and DPG meshes are adapted.  The DPG method~\cite{ Erikkson} for $Pe=10^6$ delivers \revMP{an} \revMP{$L^2$} norm error of order 0.02 for mesh size of the order 2000. The iGRM method for mesh dimension $20\times 10$ delivers \revMP{$L^2$} norm error 0.02 on mesh size 1312 with a residual space of (3,1).  Summing up, for $Pe=10^6$, iGRM delivers similar quality solutions on smaller grids. We achieve this by using smooth functions with higher continuity approximations. Additionally, the iGRM implementation is simpler since it does not require breaking the spaces. Nevertheless, the present solution strategy for iGRM is limited to tensor-product meshes. }

\revcc{Comparing the iGRM with SUPG, we conclude that iGRM converges to similar quality solutions than the SUPG method. The computational cost is higher for iGRM, but the iGRM method does not require the determination of problem-specific parameters.}

\begin{table*}[htp]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\includegraphics[scale=0.5]{cross_1.pdf} & \includegraphics[scale=0.5]{cross_3.pdf} &
\includegraphics[scale=0.5]{cross_5.pdf} \\
iteration 1 [0  0.5  1]   &  iteration 3  [0  0.5  0.75  0.875  1]  & iteration 5 [0  0.5  0.75  0.875 \\ & &  0.9375  0.96875  1]  \\
\hline
\includegraphics[scale=0.5]{cross_7.pdf} & \includegraphics[scale=0.5]{cross_9.pdf} &
\includegraphics[scale=0.5]{cross_11.pdf} \\
iteration 7  [0  0.5  0.75  0.875  & iteration 9  [0  0.5  0.75  0.875  & iteration 11 [0  0.5  0.75  0.875  \\
 0.9375  0.96875  0.984375  1]  & 0.9375  0.96875  ...  & 0.9375  0.96875  ... \\
                                                & 0.998046875  1]                 &  0.998046875 0.9990234375  \\
                                                &                                           & 0.9995117188  1] \\
\hline
\includegraphics[scale=0.5]{cross_13.pdf} & \includegraphics[scale=0.5]{cross_15.pdf} &
\includegraphics[scale=0.5]{cross_17.pdf} \\
iteration 13 [0  0.5  0.75  0.875   & iteration 15 [0  0.5  0.75  0.875  & iteration 17 [0  0.5  0.75  0.875  \\
0.9375  0.96875  ...  &  0.9375  0.96875  ... &  0.9375  0.96875  ... \\
  0.9998779297  1] &   0.9998779297  0.9999389648 &   0.9998779297  0.9999389648 \\
					  &     0.9999694824  1]  &   0.9999694824  0.9999847412 \\
					  &                                &    0.9999923706  1] \\
\hline
\end{tabular}
\end{center}
\caption{\revMP{Solutions to the Erikkson-Johnsson problem by using iGRM and SUPG methods for Pe=1,000,000 over a sequence of grids. We report the knot vector below each picture.}}
\label{fig:Crosses1}
\end{table*}

\begin{table*}[htp]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\includegraphics[scale=0.5]{zoom_18.pdf} & \includegraphics[scale=0.5]{zoom_19.pdf} &
\includegraphics[scale=0.5]{zoom_20.pdf} \\
iteration 18 [0  0.5  0.75 &  iteration 19 [0  0.5  0.75 & iteration 20 [0  0.5  0.75 \\
 0.875  0.9375  0.96875 ... &  0.875  0.9375  0.96875 ... &  0.875  0.9375  0.96875 ... \\
0.9999961853  1]  &   0.9999980927  1] & 0.9999980927 0.9999990463  1] \\
\hline
\includegraphics[scale=0.5]{zoom_21.pdf} & \includegraphics[scale=0.5]{zoom_22.pdf} &
\includegraphics[scale=0.5]{zoom_23.pdf} \\
iteration 21  [0  0.5  0.75  0.875  & iteration 22  [0  0.5  0.75  0.875  & iteration 23 [0  0.5  0.75  0.875  \\
0.9375 0.96875    ... & 0.9375 0.96875    ... &  0.9375 0.96875    ... \\
0.9999990463  0.9999995232  1]  & 0.9999990463  0.9999995232  & 0.9999990463  0.9999995232 \\
							& 0.9999997616  1]   & 0.9999997616  0.9999998808  1] \\
\hline
\includegraphics[scale=0.5]{zoom_24.pdf} & \includegraphics[scale=0.5]{zoom_25.pdf} &
\includegraphics[scale=0.5]{zoom_26.pdf} \\
iteration 21  [0  0.5  0.75  0.875  & iteration 22  [0  0.5  0.75  0.875  & iteration 23 [0  0.5  0.75  0.875  \\
0.9375 0.96875    ... & 0.9375 0.96875    ... &  0.9375 0.96875    ... \\
0.9999999404  1]                    & 0.9999999404  0.9999999702  1] & 0.9999999404  0.9999999702 \\								 &                                                            &  0.9999999851  1] \\
\hline
\end{tabular}
\end{center}
\caption{\revMP{Zoom to the right corner of the solutions to the Erikkson-Johnsson problem by using iGRM and SUPG methods for Pe=1,000,000 over a sequence of adapted grids. We report the knot vector below each picture.}}
\label{fig:Crosses2}
\end{table*}

\begin{figure}
\centering
\includegraphics[scale=0.45]{igrm.pdf}
\caption{\revcc{Solution to the Erikkson-Johnson problem on the mesh from iteration 20.}}
\label{fig:Crosses3}
\end{figure}

\begin{table*}[htp]
\begin{center}
\begin{tabular}{cccc|ccc}
& \multicolumn{3}{c}{ SUPG (2,1) } & \multicolumn{3}{c}{ iGRM (2,1) (3,1) } \\
iteration & \#NRDOF & L2 & H1  & \#NRDOF & L2 & H1  \\
\hline
1 & 24 & 44.33 & 56.37 & 84 & 53.59 & 68.00  \\
2 & 30 & 33.79 & 56.29 & 110 & 44.00 & 66.03  \\
3 & 36 & 24.74 & 66.30 & 136 & 31.86 & 90.46  \\
4 & 42 & 17.75 & 87.73 & 162 & 20.31 & 143.38  \\
5 & 48 & 12.63 & 121.57 & 188 & 12.42 & 227.77 \\
6 & 54 & 8.96 & 170.93 & 214 & 7.36 & 345.20 \\
7 & 60 & 6.34 & 241.31 & 240 & 4.49 & 506.91 \\
8 & 66 & 4.49 & 341.08 & 266 & 2.85 & 731.00 \\
9 & 72 & 3.18 & 482.28 & 292 & 1.89 & 1044.11 \\
10 & 78 & 2.26 & 681.99 & 318 & 1.31 & 1484.00 \\
11 & 84 & 1.61 & 964.41 & 344 & 0.94 & 2103.96 \\
12 & 90 & 0.70 & 1363.73 & 370 & 0.70 & 2979.06 \\
13 & 96 & 0.84 & 1928.06 & 396 & 0.55 & 4026.87 \\
14 & 102 & 0.63 & 1623.21 & 422 & 0.45 & 353.16 \\
15 & 108 & 0.49 & 104.06 & 448 & 0.39 & 76.60 \\
16 & 114 & 0.39 & 77.93 & 474 & 0.35 & 69.25 \\
17 & 120 & 0.34 & 78.38 & 500 & 0.34 & 58.28 \\
18 & 126 & 0.31 & 68.11 & 526 & 0.34 & 34.03 \\
19 & 132 & 0.30 & 48.56 & 552 & 0.34 & 12.57 \\
20 & 138 & 0.30 & 27.56 & 578 & 0.34 & 3.97 \\
21 & 144 & 0.30 & 9.59 & 604 & 0.34 & 2.45 \\
22 & 150 & 0.30 & 4.40 & 630 & 0.34 & 2.35 \\
23 & 156 & 0.30 & 2.69 & 656 & 0.34 & 2.35 \\
24 & 162 & 0.30 & 2.85 & 682 & 0.34 & 2.35 \\
25 & 168 & 0.30 & 2.85 & 708 & 0.34 & 2.35 \\
\hline
\end{tabular}
\end{center}
\caption{\revcc{Convergence of SUPG and iGRM on grids refined from \revMP{a} $2\times 2$ uniform grid, for the Erikkson-Johnson problem with $Pe=10^6$, for (2,1) B-splines. Here $\#NDOF$ is the total number of degrees of freedom, and \revMP{$L^2$} and \revMP{$H^1$} are the exact norm errors. We use $\eta=0.0001$.}}
\label{tab:iGRMSUPG_E}
\end{table*}

\revcc{We also investigate the convergence and the execution times of the iterative solver. We show in Table \ref{tab:CG} the dependence of the number of iterations of the inner and outer loops on the parameter $\eta$ from the definition of the inner product. We use the Erikkson-Johnson problem as a reference. We \revMP{implement} the iGRM code in the IGA-ADS software \cite{ED4}, which automatically uses all cores of the machine. We run the experiments on Intel i7 processor with 2.7GHz with 8 cores and 16 GB of RAM.}

\begin{table*}[htp]
\begin{center}
\begin{tabular}{ccccccc}
iteration & $\eta$ & \# outer & \# inner &  L2 & H1 & time [ms] \\
\hline
1 & 0.1 & $>100$ & $>100$ & 12.76 & 225.88 & 37 \\
1 & 0.01 & 10 & 32 & 24.95 & 166.275 & 12 \\
1 & 0.001 & 4 & 18 & 49.86 & 73.11 & 6 \\
1 & 0.0001 & 3 & 16 & 53.59 & 68.00 & 5 \\
1 & 0.00001 & 2 & 12 & 58.90 & 66.36 & 4 \\
1 & 0.000001 & 2 & 11 & 59.02 & 66.45 & 4 \\
1 & 0.0000001 & 2 & 11 & 59.02 & 66.46 & 4 \\
1 & 0.00000001 & 2 & 11 & 59.02 & 66.46 & 4 \\
\hline
20 & 0.1 & $>100$ & $>100$ & 0.34 & 42.96 & 132 \\
20 & 0.01 & $>100$ & $>100$ & 0.34 & 8.24 &103 \\
20 & 0.001 & 5 & 69 & 0.34 & 4.07 & 30 \\
20 & 0.0001 & 2 & 51 & 0.34 & 3.97 & 21\\
20 & 0.00001 & 2 & 63 & 0.34 & 3.97 & 26 \\
20 & 0.000001 & 2 & 65 & 0.34 & 3.97 & 23 \\
20 & 0.0000001 & 2 & 81 & 0.59 & 4.05 & 31 \\
20 & 0.00000001 & 2 & 85 & 4.73 & 8.91 & 34 \\
\hline
\end{tabular}
\end{center}
\caption{Convergence of the inner (CG) and outer (corrections) loops of the iterative solver for different $\eta$ parameter from the weighted norm. We use the first and the last (number 20) adaptive grids for the Erikkson-Johnson problem with $Pe=10^6$, for trial (2,1) and test (3,1) B-splines. Here \revMP{$L^2$} and \revMP{$H^1$} are the exact norm errors.}
\label{tab:CG}
\end{table*}

\subsection{Vortical wind problem}

We now solve a vortical flow with the advection-dominated diffusion equations
over the rectangular domain $\Omega=(0,1)\times(-1,1)$, with zero right-hand side $f=0$,
the advection vector $\beta(x,y)=\left(\beta_x(x,y),\beta_y(x,y)\right)=\left(-y,x\right)$. This velocity field is modeling a rotating flow.
We introduce
$\Gamma_{1}=\{(x,y):x=0, 0.5\leq y \leq 1.0\}$,
$\Gamma_{2}=\{(x,y):x=0, 0.0\leq y \leq 0.5\}$\revMP{.}
The Dirichlet boundary conditions are
\begin{eqnarray}
g = \frac{1}{2}\left(\tanh \left(\left(|y|-0.35\right)\frac{b}{\epsilon}\right)+1\right), \textrm{ for }x\in\Gamma_2 \nonumber \\
g = \frac{1}{2}\left(0.65-\tanh \left(\left(|y|\right)\frac{b}{\epsilon}\right)+1\right), \textrm{ for }x\in\Gamma_1 \\
g=0, \textrm{ for }x\in\Gamma \setminus \Gamma_1\cup\Gamma_2
\label{eq:Problem4_bc}
\end{eqnarray}

We use the iGRM setup (\ref{eq:resmin}) with the preconditioned CG solver described in Section 4. We use $128\times 128$ mesh with solution (2,1) residual (2,0). We use a P\'eclet number $Pe=10^6$. The numerical results are summarized in Figures~\ref{fig:Problem4a}-\ref{fig:Problem4c}.

There are some overshoots and undershoots in the cross-sections of the mesh. They can be removed by performing mesh refinements. They can be also removed by using penalty methods~\cite{ yzbeta1,yzbeta2}.

\begin{figure}
\centering
\includegraphics[scale=0.45]{128_21_20_bars.pdf}
\caption{Solution to the circular wind problem on a mesh of $128\times
  128$ elements with trial (2,1), test (2,0), for P\'eclet number $Pe=10^6$, wind force $b=1$. We also include two planes (cross-sections) whose solutions are presented in Figures~\ref{fig:Problem4b}-\ref{fig:Problem4c}.}
\label{fig:Problem4a}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.5]{128_21_20_hor.pdf}
\caption{Horizontal cross-section at $x=0$ through the solution to the circular wind problem on a mesh of $128\times 128$ elements with trial (2,1), test (2,0), for P\'eclet number $Pe=10^6$, wind force $b=1$.}
\label{fig:Problem4b}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.5]{128_21_20_ver.pdf}
\caption{Vertical cross-section at $y=0.2$ through the solution to the circular wind problem on a mesh of $128\times 128$ elements with trial (2,1), test (2,0), for P\'eclet number $Pe=10^6$, wind force $b=1$.}
\label{fig:Problem4c}
\end{figure}

\begin{remark}
The number of iterations of the conjugate gradient solver grows with the problem size, as presented in Table \ref{tab:CG}. We will explore possible solutions to this issue in future work.
\revMP{The computational cost of the solver grows like ${\cal O}(Nk)$ where $N$ is the number of degrees of freedom, and $k$ is the number of iterations.}
\end{remark}

\section{Conclusions}
\label{sec:conc}

\revcc{We introduce a stabilized isogeometric method that uses residual minimization in a dual norm.  We design the method to achieve good solution properties. The method exploits the Kronecker product structure of the computational problem to deliver fast solutions. The solution space in our scheme uses maximum continuity B-splines. To accelerate the solution of the algebraic scheme, we introduce a fast solver for the Gramm matrix, but without a Schur complement preconditioner.  We call our method isogeometric residual minimization (iGRM) with direction splitting preconditioner.  We verify the accuracy of the solution on four stationary problems.  In this method, the diffusion and advection coefficient functions can be arbitrary.  Our future work will extend this method to other problems, such as the Stokes~\cite{ evans2013isogeometric} and Maxwell problems~\cite{ maxwell1, maxwell2,maxwell3}, the development of the method for time-dependent problems~\cite{ timedep}, as well as the development of the parallel software dedicated to the simulations of different non-stationary problems with the iGRM method.  We will seek to find a proper preconditioner for the CG problem as well as an alternative method to speed up the solution of the saddle-point problem that the residual minimization delivers, e.g., based on~\cite{ Method1, Method2}.}

\subsection*{Acknowledgments}

The National Science Centre, Poland,  partially supported this work, grant no. 2017/26/M/ ST1/ 00281.  This publication was also made possible in part by the CSIRO Professorial Chair in Computational Geoscience at Curtin University and the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia. Additional support was provided by the European Union's Horizon 2020 Research and Innovation Program of the Marie Skłodowska-Curie grant agreement No. 777778. At Curtin University, support was provided by The Institute for Geoscience Research (TIGeR) and by the Curtin Institute for Computation.

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