A Finite Element Method by Patch Reconstruction for the Quad-Curl Problem Using Mixed Formulations

TL;DR

This paper introduces a high-order patch reconstruction finite element method for solving the quad-curl problem with mixed formulations, achieving optimal convergence rates in energy norms and verified through numerical experiments.

Contribution

It presents a novel high-order reconstructed discontinuous approximation method for the quad-curl problem using mixed formulations with optimal convergence analysis.

Findings

Optimal convergence rate under the energy norm.

Suboptimal $L^2$ convergence demonstrated.

Numerical results confirm theoretical predictions.

Abstract

We develop a high order reconstructed discontinuous approximation (RDA) method for solving a mixed formulation of the quad-curl problem in two and three dimensions. This mixed formulation is established by adding an auxiliary variable to control the divergence of the field. The approximation space for the original variables is constructed by patch reconstruction with exactly one degree of freedom per element in each dimension and the auxiliary variable is approximated by the piecewise constant space. We prove the optimal convergence rate under the energy norm and also suboptimal $L^2$ convergence using a duality approach. Numerical results are provided to verify the theoretical analysis.