HDG Methods for the two-dimensional Vector Laplacian
This work provides incremental improvements in numerical methods for partial differential equations, specifically targeting computational physics or engineering applications involving vector fields.
The paper tackles the problem of solving the two-dimensional vector Laplacian equation with various boundary conditions by introducing new hybridizable discontinuous Galerkin (HDG) methods, achieving optimal convergence rates of k+1 for the electric field and k+1/2 for auxiliary variables in L^2 norm.
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.