GPU-based matrix-free multigrid methods of stable generalized finite element methods for interface problems

It is challenging for generalized/extended finite element methods (G/XFEMs) to develop fast solvers for linear systems because (a) their stiffness matrices are not standard FE matrix, which are composed of FE, enrichment, and intersection parts, and (b) the trial spaces of fine and coarse meshes do not possess nested structures. GPU accelerated multigrid (MG) techniques for the G/XFEM are thus rarely realized. This study proposes a GPU-based matrix-free MG for stable GFEM (SGFEM, a stable version of G/XFEM) for interface problems. A linear system of SGFEM is transformed into two small systems using a Schur complement scheme, where the major computational expense is concentrated on solving one small system with matrix (Formula presented). M is a FE matrix, and P is a small perturbation. By analyzing the effect of P[jls-end-space/], we find that the conditioning of (Formula presented) is of the same order as that of M[jls-end-space/]. This motivates us to design the MG of (Formula presented) using the same prolongation and restriction operations for the FE matrix M[jls-end-space/]. Such a scheme can also enable GPU-accelerated techniques. To reduce the GPU memory, we develop matrix-free MG (MFMG) methods for (Formula presented). Different from the MF operations of FEM, the MF algorithm of (Formula presented) needs to incorporate the effect of P[jls-end-space/]. To overcome it, we design novel MF algorithms for matrix–vector multiplications and MG-smoothers based on M and P[jls-end-space/]. The proposed MFMG maintains optimal convergence rates of SGFEM. Most importantly, the computational complexity of MFMG is not increased essentially, which is of same order as that of standard FEs. Comparisons with conventional GPU solvers are made to demonstrate advantages of the proposed MFMG in computational time and memory overhead, including NVIDIA AMGX.