Convergence of an IP DG Method for the Quad-Curl Problem

Abstract

This work analyzes revises the interior penalty (IP) discontinuous Galerkin (DG) method imposed in [Chen, G., Qiu, W., \& Xu, L. (2021). Analysis of an interior penalty DG method for the quad-curl problem. IMA Journal of Numerical Analysis, 41(4), 2990-3023.] for the quad-curl problem in a nonconvex polyhedral domain, while introducing a piecewise constant coefficient matrix. We derive two main results: Under minimal regularity assumptions, we prove that the numerical solutions converge strongly to the true solution in the $H(\text{curl})\times H^1(\hat{I}\text{c◯})$ norm. Under slightly higher regularity, we establish the optimal estimate of the convergence rate depending on the regularity of the solution. These two results, serving as a complement to the existing literature, completely answer how the concerned IP DG method performs on quad-curl problems with low regularity.

33 pages