Hybridizable discontinuous Galerkin method for curved domains

Abstract

In this work we present a technique to numerically solve partial differential equations (PDE’s) defined in general domains ­. It basically consists in approximating the domain ­ by polyhedral subdomains Dh and suitably defining extensions of the solution from Dh to ­. More precisely, we solve the PDE in Dh by using a numerical method for polyhedral domains. In order to do that, the boundary condition is transferred from ¡ := ∂­ to ¡h, the boundary of Dh, by integrating the gradient of the scalar variable along a path. That is why, in principle, any numerical method that provides an accurate approximation of the gradient can be used. In this work we consider a hybridizable discontinuous Galerkin (HDG) method. This technique has two main advantages over other methods in the literature. First of all, it only requires the distance between ¡ and ¡h to be of the order of the meshsize. This allows us to easily mesh the computational domain. Moreover, high degree polynomial approximations can be used and still obtain optimal orders of convergence even though ¡h is “far” from ¡. We numerically explore this approach by considering three types of steady-state equations. As starting point, we deal with Dirichlet boundary problems for second order elliptic equations. For this problem we fully explain how to properly transfer the boundary condition and how to define the paths, as well. We then apply this technique to exterior diffusion problems. Herein, the HDG method is used for solving the so-called interior problem on a bounded region whereas a boundary element method (BEM) is used for solving the problem exterior to that region. Both methods are coupled at the smooth interface that divides the two regions. Finally, we consider convectiondiffusion problems where we explore how the magnitude of the convective field affects the performance of our method.