Unified geometric multigrid algorithm for hybridized high-order finite element methods

TL;DR

This paper introduces a unified geometric multigrid algorithm for hybridized high-order finite element methods, leveraging DtN maps to efficiently solve elliptic PDEs on unstructured meshes without parameter upscaling.

Contribution

The paper presents a physics-based, energy-preserving intergrid transfer operator approach that applies to various hybridized finite element methods, simplifying multigrid implementation and improving scalability.

Findings

Effective multigrid convergence on complex elliptic problems

Robust performance across different hybridized methods

Scalability with unstructured meshes and heterogeneous media

Abstract

We consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. The proposed unified approach is applicable to any locally conservative hybridized finite element method including multinumerics with different hybridized methods in different parts of the domain. For these methods, the linear system involves only the unknowns residing on the mesh skeleton, and constructing intergrid transfer operators is therefore not trivial. The key to our geometric multigrid algorithm is the physics-based energy-preserving intergrid transfer operators which depend only on the fine scale DtN maps. Thanks to these operators, we completely avoid upscaling of parameters and no information regarding subgrid physics is explicitly required on coarse meshes. Moreover, our algorithm is agglomeration-based and can straightforwardly handle unstructured meshes. We perform extensive numerical studies with hybridized mixed methods, hybridized discontinuous Galerkin method, weak Galerkin method, and a hybridized version of interior penalty discontinuous Galerkin methods on a range of elliptic problems including subsurface flow through highly heterogeneous porous media. We compare the performance of different smoothers and analyze the effect of stabilization parameters on the scalability of the multigrid algorithm.