Local Fourier analysis of multigrid for hybridized and embedded discontinuous Galerkin methods

TL;DR

This paper analyzes the efficiency of multigrid methods with Jacobi and Vanka relaxation for hybridized and embedded discontinuous Galerkin discretizations of the Laplacian, using local Fourier analysis to compare their performance.

Contribution

It provides a local Fourier analysis of multigrid methods for EDG and HDG discretizations, demonstrating near-equivalence with continuous Galerkin and superior performance of EDG.

Findings

Multigrid with EDG discretization is nearly as efficient as with continuous Galerkin.

Multigrid applied to EDG outperforms that applied to HDG.

Numerical results confirm the LFA predictions.

Abstract

In this paper we present a geometric multigrid method with Jacobi and Vanka relaxation for hybridized and embedded discontinuous Galerkin discretizations of the Laplacian. We present a local Fourier analysis (LFA) of the two-grid error-propagation operator and show that the multigrid method applied to an embedded discontinuous Galerkin (EDG) discretization is almost as efficient as when applied to a continuous Galerkin discretization. We furthermore show that multigrid applied to an EDG discretization outperforms multigrid applied to a hybridized discontinuous Galerkin (HDG) discretization. Numerical examples verify our LFA predictions.