Multiscale Sub-grid Correction Method for Time-Harmonic High-Frequency Elastodynamics with Wavenumber Explicit Bounds

TL;DR

This paper introduces a multiscale sub-grid correction method for high-frequency elastodynamics that remains pollution free and is supported by polynomial-in-wavenumber stability bounds, validated through numerical experiments.

Contribution

It develops a pollution-free Petrov-Galerkin multiscale method with wavenumber-explicit bounds, avoiding geometric constraints used in previous approaches.

Findings

Method eliminates pollution in natural and oversampling regimes.

Numerical results show improved accuracy over standard finite elements.

Establishes polynomial-in-k bounds for elastodynamics stability constants.

Abstract

The simulation of the elastodynamics equations at high-frequency suffers from the well known pollution effect. We present a Petrov--Galerkin multiscale sub-grid correction method that remains pollution free in natural resolution and oversampling regimes. This is accomplished by generating corrections to coarse-grid spaces with supports determined by oversampling lengths related to the $\log(k)$, $k$ being the wavenumber. Key to this method are polynomial-in-$k$ bounds for stability constants and related inf-sup constants. To this end, we establish polynomial-in-$k$ bounds for the elastodynamics stability constants in general Lipschitz domains with radiation boundary conditions in $\mathbb{R}^3$. Previous methods relied on variational techniques, Rellich identities, and geometric constraints. In the context of elastodynamics, these suffer from the need to hypothesize a Korn's inequality on the boundary. The methods in this work are based on boundary integral operators and estimation of Green's function's derivatives dependence on $k$ and do not require this extra hypothesis. We also implemented numerical examples in two and three dimensions to show the method eliminates pollution in the natural resolution and oversampling regimes, as well as performs well when compared to standard Lagrange finite elements.