On a two-dimensional discontinuous Galerkin discretisation with embedded Dirichlet boundary condition
In this paper we introduce a discretisation 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 discretisation in the cell in which the (embedded) Dirichlet boundary cannot follow the gridlines of the orthogonal grid.The DG-discretisation aims at a high order of accuracy. We discretize with tensor products of cubic polynomials. By construction, such a DG discretisation is fourth order consistent, both in the interior and at the boundaries. By experiments we show fourth order convergence in the presence of a curved Dirichlet boundary. Stability is proved for the one-dimensional Poisson equation with an embedded boundary condition.To illustrate the possibilities of our DG-discretisation, we solve a convection dominated boundary value problem on a regular rectangular grid with a circular embedded boundary condition . We show how accurately the boundary layer with a complex structure can be captured by means of piece-wise cubic polynomials. The example shows that the embedded boundary treatment is effective.