Analysis of injection operators in multigrid solvers for hybridized discontinuous Galerkin methods

TL;DR

This paper proves uniform convergence of geometric multigrid V-cycle for HDG methods under new assumptions on injection operators, including local operators, supported by theoretical analysis and numerical validation.

Contribution

Introduces new assumptions on injection operators for multigrid convergence in HDG methods, including local operators, with comprehensive theoretical and numerical validation.

Findings

Proves uniform convergence of multigrid V-cycle for HDG methods.

Validates theoretical results with numerical experiments.

Applies to multiple hybridized mixed finite element methods.

Abstract

Uniform convergence of the geometric multigrid V-cycle is proven for HDG methods with a new set of assumptions on the injection operators from coarser to finer meshes. The scheme involves standard smoothers and local solvers which are bounded, convergent, and consistent. Elliptic regularity is used in the proofs. The new assumptions admit injection operators local to a single coarse grid cell. Examples for admissible injection operators are given. The analysis applies to the hybridized local discontinuous Galerkin method, hybridized Raviart-Thomas, and hybridized Brezzi-Douglas-Marini mixed element methods. Numerical experiments are provided to confirm the theoretical results.