What are all these functions in FEM? Shape function vs Basis Function vs Trial Function vs Test Function vs Interpolation Function
I am a novice in FEM. I have some experience with FDM which was pretty straight forward. Since I have a confusion with a number of concepts, I will try to break them down by writing down what I have understood and what the confusion. Sorry if the question is too long.
(Firstly, all the FEM textbooks start off by saying its a method used to solve PDEs, etc and then directly goes on to using linear spring as an element and derivation of stiffness matrix without mentioning anything about the said PDEs. This is really confusing.)
1) If I understand correctly, the solution to the given problem is talked about either in terms of within an element or the solution throughout the entire computational domain(the actual solution). The Shape functions come into play with-in an element. 232
It approximates the "shape" of the solution within that element. It interpolates the solution values at node points throughout an element and it is what defines an element.
a)Are shape functions and interpolation functions the same thing?
b) Are there one shape function for each of the node points?
c)Are the value of the shape function unity for a particular node point and zero at other node points? (I am asking b and c because there is a confusion with Shape function and Basis function, with some explanation stating basis function has a value of unity at node points. Ref: https://www.comsol.com/multiphysics/finite-element-method)
2) Basis functions come to play when we are stating the solution throughout the computational domain as a linear combination of the basis function. This states the solution at the global level (entire computational domain). How is it brought down to the element level from the global level? ie how are basis functions and the shape functions related? (because once the explanation of basis function begins there is no mention of shape funcion, which makes me suspect the basis function that was used at the global level in the weighted residual method is same as the shape function at the element level in the FEM.)
a) Are basis functions and trial functions the same thing?
b)Are trial function and test functions the same thing?
c) What is the difference between basis function and shape function?
3) Just to get clarity between FDM and FEM: In FDM we replace the derivatives with a finite difference equation ie we approximate the given differential equation itself but in FEM we try to approximate the solution to the differential equation. Is my understanding correct? 232