A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity

Abstract:

We present the first a priori error analysis of a technique that allows us to numerically solve steady-state diffusion problems defined on curved domains $\Omega$ by using finite element methods defined in polyhedral subdomains $\mathsf {D}_h\subset \Omega$. For a wide variety of hybridizable discontinuous Galerkin and mixed methods, we prove that the order of convergence in the $L^2$-norm of the approximate flux and scalar unknowns is optimal as long as the distance between the boundary of the original domain $\Gamma$ and that of the computational domain $\Gamma _h$ is of order $h$. We also prove that the $L^2$-norm of a projection of the error of the scalar variable superconverges with a full additional order when the distance between $\Gamma$ and $\Gamma _h$ is of order $h^{5/4}$ but with only half an additional order when such a distance is of order $h$. Finally, we present numerical experiments confirming the theoretical results and showing that even when the distance between $\Gamma$ and $\Gamma _h$ is of order $h$, the above-mentioned projection of the error of the scalar variable can still superconverge with a full additional order.