Matrix free finite elements method

Question

I am Computer Scientist and now I am interested in matrix multiplication on GPUs. My research are focused on matrix free finite elements method where I multiply sparse matrix. Sparse matrix could multiply regular or matrix free. In general based on special coordinate function. I have a few general question: How popular is this method? Does any another name exist for this method? I am also looking for books and article concentrate on finite element method especially matrix free multiplication and I consider about general books and article. Because many article based on rather complicated examples like conjugate gradient method or something different.

Answer 1

Matrix-free finite elements are relatively well-known. For explicit methods for transient problems, this involves applying the finite element matrix using small reference matrices and geometry-specific transformations.

For implicit problems, this is usually done in conjunction with an iterative solver such as CG, GMRES, etc. Note that this typically also requires a preconditioner, many of which may not map well directly to GPUs. For nonlinear implicit problems, this may also be paired with a matrix-free approximation of the Jacobian in a linearization (see for example Ben Kirk's thesis for an application of matrix-free implicit methods in CFD).

For GPU-specific implementations of explicit methods, there is ample literature on matrix-free implementations of finite element (specifically Discontinuous Galerkin) methods; see, for example, Hesthaven/Warburton (not GPU specific, but implementation is similar) and Klockner et al.. For some literature on the GPU implementation of matrix-free implicit solvers, see Remacle, Gandham, Warburton, where they solve the heat equation using a two-grid overlapping additive Schwarz preconditioner.