An $hp$ Error Analysis of HDG for Linear Fluid-Structure Interaction

TL;DR

This paper develops an $hp$-error analysis for a hybridizable discontinuous Galerkin method applied to linear fluid-structure interaction problems, demonstrating stability, convergence, and validating through numerical experiments.

Contribution

It introduces an $hp$-convergence analysis for HDG methods in FSI problems, including stability and error estimates, with comprehensive numerical validation.

Findings

The method is energy stable and well-posed.

Convergence rates are established for semi-discrete and fully discrete schemes.

Numerical experiments confirm theoretical error estimates.

Abstract

A variational formulation based on velocity and stress is developed for linear fluid-structure interaction (FSI) problems. The well-posedness and energy stability of this formulation are established. To discretize the problem, a hybridizable discontinuous Galerkin method is employed. An $hp$-convergence analysis is performed for the resulting semi-discrete scheme. The temporal discretization is achieved via the Crank-Nicolson method, and the convergence properties of the fully discrete scheme are examined. Numerical experiments validate the theoretical results, confirming the effectiveness and accuracy of the proposed method.