A parallel and adaptive hybridized discontinuous Galerkin method for anisotropic nonhomogeneous diffusion

Abstract

The diffusion equation in anisotropic and nonhomogeneous media arises in the study of flow through porous media with sharp material interfaces. We discuss the solution of this problem by a hybrid discontinuous Galerkin (HDG) method. The method can be applied in three steps. First, we use a condensation technique to derive the scalar variable and the flux inside each element in terms of the numerical trace on the faces of that element. Then we form a global system of equations to solve for these numerical traces. We then solve the equation inside each element for the internal unknowns using the obtained numerical traces in the global solve step. Similar to other DG variants, HDG is a locally conservative method, and a noticeable share of calculations are performed independently within each element. In a mesh with pth order elements (p≥0), this method gives p+1 order of accuracy for smooth solutions for both the scalar variable and the flux. Moreover, by using a simple post-processing technique, one can reach an accuracy of order p+2 for the scalar variable. To be able to handle problems with sharp material discontinuities, we use an adaptive refinement strategy, with octree grid structure. Hence, we avoid a larger global system which would arise from a fine uniform grid. In this process, we refine those elements with highest gradient of flux, at two sides of their faces. We have also utilized shared and distributed memory parallelism to enhance the performance of the method. The method is implemented using different modules of deal.II (Bangerth and Kanschat, 1999), PETSc (Balay et al., 2015), p4est (Burstedde et al., 2011) and Hypre (Falgout and Yang, 2002). To demonstrate the accuracy, efficiency, scalability, and flexibility of the method, several two and three dimensional numerical experiments are studied.