An hp-version interior penalty discontinuous Galerkin method for the quad-curl eigenvalue problem

TL;DR

This paper introduces an hp-version interior penalty discontinuous Galerkin method for solving the quad-curl eigenvalue problem, providing theoretical analysis and demonstrating optimal convergence rates through numerical experiments.

Contribution

It develops a novel hp-version IPDG method for the quad-curl eigenvalue problem, including proof of well-posedness, error estimates, and convergence analysis on nonconforming meshes.

Findings

Optimal h-version convergence rate observed.

Exponential p-version convergence rate demonstrated.

Method performs well on conforming and nonconforming meshes.

Abstract

An hp-version interior penalty discontinuous Galerkin (IPDG) method under nonconforming meshes is proposed to solve the quad-curl eigenvalue problem. We prove well-posedness of the numerical scheme for the quad-curl equation and then derive an error estimate in a mesh-dependent norm, which is optimal with respect to h but has different p-version error bounds under conforming and nonconforming tetrahedron meshes. The hp-version discrete compactness of the DG space is established for the convergence proof. The performance of the method is demonstrated by numerical experiments using conforming/nonconforming meshes and h-version/p-version refinement. The optimal h-version convergence rate and the exponential p-version convergence rate are observed.