A high-order immersed boundary discontinuous-Galerkin method for Poisson's equation with discontinuous coefficients and singular sources

We adopt a numerical method to solve Poisson's equation on a fixed grid with embedded boundary conditions, where we put a special focus on the accurate representation of the normal gradient on the boundary. The lack of accuracy in the gradient evaluation on the boundary is a common issue with low-order embedded boundary methods. Whereas a direct evaluation of the gradient is preferable, one typically uses post-processing techniques to improve the quality of the gradient. Here, we adopt a new method based on the discontinuous-Galerkin (DG) finite element method, inspired by the recent work of [A.J. Lew and G.C. Buscaglia. A discontinuous-Galerkin-based immersed boundary method. International Journal for Numerical Methods in Engineering, 76:427-454, 2008]. The method has been enhanced in two aspects: firstly, we approximate the boundary shape locally by higher-order geometric primitives. Secondly, we employ higher-order shape functions within intersected elements. These are derived for the various geometric features of the boundary based on analytical solutions of the underlying partial differential equation. The development includes three basic geometric features in two dimensions for the solution of Poisson's equation: a straight boundary, a circular boundary, and a boundary with a discontinuity. We demonstrate the performance of the method via analytical benchmark examples with a smooth circular boundary as well as in the presence of a singularity due to a re-entrant corner. Results are compared to a low-order extended finite element method as well as the DG method of [1]. We report improved accuracy of the gradient on the boundary by one order of magnitude, as well as improved convergence rates in the presence of a singular source. In principle, the method can be extended to three dimensions, more complicated boundary shapes, and other partial differential equations. Copyright (c) 2014 John Wiley & Sons, Ltd.