A feasibility study for discontinuous Galerkin discretization with embedded Dirichlet boundary condition

Abstract

In this paper we introduce a discretization of Discontinuous Galerkin (DG) type for solving 2-D second-order elliptic PDEs on a regular rectangular grid, while the boundary value problem has a curved Dirichlet boundary. According to the same principles that underlie DG-methods, we adapt the discretization in the cell in which the (embedded) Dirichlet boundary cannot follow the gridlines of the orthogonal grid. The DG-discretization aims at a high order of accuracy. We discretize with tensor products of cubic polynomials and we parameterize the embedded boundary by cubic polynomials. Then, by construction, such a DG discretization is fourth-order consistent, both in the interior and at the boundaries. The proposed discretization technique is motivated by the fact that it results in only a slight modification of DG discretization at embedded boundaries and requires a much simpler mesh generation than needed for boundary conforming meshes. In particular the method can show its use in cases where regular rectangular grids are preferred. To illustrate the possibilities of our DG-discretization, we solve a convection dominated boundary value problem on a regular rectangular grid with a circular embedded boundary condition [J. Comput. Appl. Math. 76 (1997) 277]. We show how accurately the boundary and shear layer, emanating from the curved embedded boundary, can be captured by means of the tensor product polynomials on the structured orthogonal mesh.