Exponential convergence of \(hp\)-FEM for elliptic problems in polyhedra: mixed boundary conditions and anisotropic polynomial degrees
The authors consider \(hp\)-version finite element methods on geometric meshes consisting of hexahedral elements for linear, second-order elliptic boundary value problems in axiparallel polyhedral domains. Exponential rates of convergence are proved. The results extend and generalize the ones of the authors' previous work [Math. Models Methods Appl. Sci. 25, No. 9, 1617--1661 (2015; Zbl 1322.65099)] which dealt with homogeneous Dirichlet boundary conditions and uniform isotropic polynomial degrees to mixed Dirichlet-Neumann boundary conditions and to anisotropic hexahedra. \(H^1\)-conforming quasi-interpolation operators with \(N\) degrees of freedom are constructed. Exponential consistency bounds \(\exp (-b\root 5 \of {N})\) for piecewise analytic functions with singularities at edges, vertices and interfaces of boundary conditions are proved.
second-order elliptic problems in polyhedra
scientific article; zbMATH DE number 6916111
scientific article