An adaptive composite discontinuous Galerkin method for elliptic problems on complicated domains with discontinuous coefficients

Abstract

In this paper, we introduce the Multi-Region Discontinuous Galerkin Composite Finite Element Method (MRDGCFEM) with hp-adaptivity for the discretization of second-order elliptic partial differential equations with discontinuous coefficients. This method allows for the approximation of problems posed on computational domains where the jumps in the diffusion coefficient form a micro-structure. Standard numerical methods could be used for such problems but the computational effort may be extremely high. Small enough elements to represent the underlying pattern in the diffusion coefficient have to be used. In contrast, the dimension of the underlying MRDGCFE space is independent of the complexity of the diffusion coefficient pattern. The key idea is that the jumps in the diffusion coefficient are no longer resolved by the mesh where the problem is solved; instead, the finite element basis (or shape) functions are adapted to the diffusion pattern allowing for much coarser meshes. In this paper, we employ hp-adaptivity on a series of test cases highlighting the practical application of the proposed numerical scheme.

Higher Order Composite DG approximations of Gross–Pitaevskii ground state: Benchmark results and experiments

“…This means that T h 1 is a mesh which has the granularity that is affordable to solve the problem, but not fine enough to resolve all the details of Ω or to fully represent the coefficient functions of the partial differential operator. Details of the coarsening andrefinement algorithm to construct such a sequence of meshes, for the given geometry Ω, can be found in [2,18]. In order to efficiently transition from one refinement level to the next, one needs an efficient implementation of coarsening and refinement operators.…”