HDG Methods for the two-dimensional Vector Laplacian

TL;DR

This paper develops new hybridizable discontinuous Galerkin methods for the 2D vector Laplacian, achieving optimal convergence rates for electric fields and auxiliary variables under various boundary conditions.

Contribution

The paper introduces novel HDG methods formulated on a first-order system, with three implementation variants and proven optimal convergence rates for all variables.

Findings

Electric field error converges at rate k+1 in L2 norm.

Auxiliary variables' errors converge at order k+1/2.

Numerical tests confirm theoretical convergence and optimality.

Abstract

We introduce new hybridizable discontinuous Galerkin (HDG) methods for solving the two-dimensional vector Laplacian equation under three types of boundary conditions: electric, magnetic, and Dirichlet. The method is formulated on a first-order system form of the equations, in which the rotational and divergence of the electric field are introduced as auxiliary variables. We study the well-posedness of the method and prove that, when using piecewise polynomial approximations of degree $k \geq 0$, the error in the $L^2$ norm of the electric field converges at the optimal rate of $k+1$. Additionally, we prove that the $L^2$-errors of the auxiliary variables, the rotational and divergence, converge with order $k + 1/2$. We also show that the methods can be implemented in three different forms, corresponding to three distinct hybridizations based on the choice of the globally coupled unknowns among the numerical traces defined on the mesh skeleton. Finally, we provide numerical tests that not only validate the theoretical convergence rates but also consistently showcase the optimal convergence across all variables.