A multigrid method for CutFEM and its implementation on GPU

TL;DR

This paper introduces a multigrid method tailored for unfitted finite element discretizations using CutFEM, optimized for GPU implementation, demonstrating fast convergence and moderate runtime overhead compared to fitted methods.

Contribution

It develops a multigrid solver for CutFEM with a parallel vertex-patch smoother and GPU implementation, addressing stabilization and boundary condition enforcement.

Findings

Fast convergence demonstrated in computational results.

Moderate runtime overhead compared to fitted methods.

Effective GPU implementation of the multigrid method.

Abstract

We present a multigrid method for an unfitted finite element discretization of the Dirichlet boundary value problem. The discretization employs Nitsche's method to implement the boundary condition and additional face based ghost penalties for stabilization. We apply standard intergrid operators, relying on the fact that the relevant domain of computation does not grow under mesh refinement. The smoother is a parallel implementation of the multiplicative vertex-patch smoother with inconsistent treatment of ghost penalties. Our computational results show that we obtain a fast converging method. Furthermore, runtime comparison to fitted methods show that the losses are moderate although many optimizations for Cartesian vertex patches cannot be applied on cut patches.