Towards a complete FEM-based simulation toolkit on GPUs: Unstructured Grid Finite Element Geometric Multigrid solvers with strong smoothers based on Sparse Approximate Inverses

Abstract: We describe our FE-gMG solver, a finite element geometric multi-grid approach for problems relying on unstructured grids. We augment our GPU- and multicore-oriented implementation technique based on cascades of sparse matrix-vector multiplication by applying strong smoothers. In particu-lar, we employ Sparse Approximate Inverse (SPAI) and Stabilised Approximate Inverse (SAINV) techniques. We focus on presenting the numerical efficiency of our smoothers in combination with low- and high-order finite element spaces as well as the hardware efficiency of the FE-gMG. For a representative problem and computational grids in 2D and 3D, we achieve a speedup of an average of 5 on a single GPU over a multithreaded CPU code in our benchmarks. In addition, our strong smoothers can deliver a speedup of 3.5 depending on the element space, compared to simple Jacobi smoothing. This can even be enhanced to a factor of 7 when combining the usage of Approximate Inverse-based smoothers with clever sorting of the degrees of freedom. In total the FE-gMG solver can outperform a simple, (multicore-)CPU-based multigrid by a total factor of over 40