Hierarchical multigrid approaches for the finite cell method on uniform and multi-level hp-refined grids

TL;DR

This paper introduces a hierarchical multigrid method tailored for large-scale finite cell problems, effectively handling complex geometries and high polynomial orders with simplified operators and robust smoothing techniques.

Contribution

It presents a novel multigrid scheme leveraging hierarchical basis functions and overlay meshes, with simple restriction/prolongation operators and Schwarz smoothing for efficient large-scale solutions.

Findings

Convergence rates are independent of cut cell configurations and mesh size.

The method efficiently solves problems with millions to billions of unknowns.

Numerical examples demonstrate scalability on massively parallel systems.

Abstract

This contribution presents a hierarchical multigrid approach for the solution of large-scale finite cell problems on both uniform grids and multi-level hp-discretizations. The proposed scheme leverages the hierarchical nature of the basis functions utilized in the finite cell method and the multi-level hp-method, which is attributed to the use of high-order integrated Legendre basis functions and overlay meshes, to yield a simple and elegant multigrid scheme. This simplicity is reflected in the fact that all restriction and prolongation operators reduce to binary matrices that do not need to be explicitly constructed. The coarse spaces are constructed over the different polynomial orders and refinement levels of the immersed discretization. Elementwise and patchwise additive Schwarz smoothing techniques are used to mitigate the influence of the cut cells leading to convergence rates that are independent of the cut configuration, mesh size and in certain scenarios even the polynomial order. The multigrid approach is applied to second-order problems arising from the Poisson equation and linear elasticity. A series of numerical examples demonstrate the applicability of the scheme for solving large immersed systems with multiple millions and even billions of unknowns on massively parallel machines.