Mixed finite element approximations of a singular elliptic problem based on some anisotropic and hp-adaptive curved quarter-point elements

摘要

Mixed finite element methods are applied to a Poisson problem with a singularity at a boundary point. The approximation spaces are based on quarter-point elements, the shape functions inheriting the singular behavior of their quadratic geometric maps. Two mesh scenarios are considered, by fixing some macro quarter-point elements at the coarse level, and subdividing them by mapping uniformly refined square meshes on the master element by their corresponding geometric transforms. For eight-noded coarse quadrilateral quarter-point elements, placing two mid-side nodes near the singular vertex, the radial singularity is exactly captured along element edges, and their refinements reveal shape regular curved meshes. For an improved version, using collapsed quadrilateral quarter-point elements obtained by reducing one of the quadrilateral element edges to the singular point, the radial singularity is captured inside the coarse macro elements as well. Their uniform refinement generates anisotropic meshes, grading towards the singular point. The assembly of the required H(div)-conforming approximation spaces based on these kinds of meshes are described. Results for a typical test problem demonstrate superior effectiveness of the proposed techniques for convergence acceleration, when confronted with usual affine finite elements, for h, p and hp-adaptive refinements. Especially, collapsed quarter-point elements applied to the singular problem reveal accuracy rates equivalent to standard regular contexts, of smooth solutions discretized on uniform affine meshes.