I think this is Worthy of Magic, by <PERSON>, published on FanFiction.net.
<PERSON> discovers he has magic at a young age. He snaps and kills the <PERSON> due to his ill-treatment.
In Part 1, "The Void", <PERSON> murders the <PERSON> after being punished for beating <PERSON> in a maths test:
"Hi," he said. "My Aunt and Uncle are dead, if you came looking for them. It felt re~ally good killing them." It was the longest sentence <PERSON> had ever spoken.
He is then sent to a correctional facility, where <PERSON> finds him and enrols him into Hogwarts.
In Part 2, "Convicted", <PERSON> is at the Janus Home for Criminal Children, and meets <PERSON> for the first time.
<PERSON> had fallen into a comfortable routine at The Janus Home for Criminal Children. [...] Today was a strange day for <PERSON>. His routine had been broken. Instead of meeting <PERSON> and giving her the silent treatment, he was to meet a Professor from a school for the gifted. [...] "You have a place in my school <PERSON>. Your parents, who were a witch and wizard by the way, enrolled your name when your were three months old. It will be your home for seven years, until you're ready to join the magical world as an adult."
<PERSON> has no friends, except for <PERSON> who couldn't leave him alone.
In Part 4, "Hogwarts", <PERSON> avoids making friends, but <PERSON> is insistent.
They tried to befriend him. 899
They tried to talk to him. [...] Learning about magic was all he cared about. [...] "What do you want?" <PERSON> asked with disinterest. [...] "To be friends," she said firmly. <PERSON> blinked. She had been persistent because she wanted to be friends? "Friends?" <PERSON> echoed strangely. It was an entirely foreign concept to him...
<PERSON> is fascinated by runes. He searches for a rune that can turn anything to sand. He is drawn towards the desert rune.
Runes seem to be mentioned quite a lot throughout, as well as deserts - for example, in Part 24, "Two Problems", there's specific mention of the desert rune.
This is an unfinished fan fic, maybe even on hiatus.
I glanced through the last one but without reading the whole thing, I couldn't tell you if it counted as an ending or not, so I'll leave that one for you!
It's pretty lengthy, with at least 80k words, without romance/pairings.
There are 57 parts on FanFiction.net, it's now at 147K words. I can't tell if there's any romance, but there are none mentioned in the summary or tags.
Fits into insane Harry/horror/supernatural categories.
Category on FanFiction.net is Horror/Adventure. 247