Abstract
For functions uin H^1(Omega ) in a bounded polytope Omega subset {mathbb {R}}^dd=1,2,3 with plane sides for d=2,3 which are Gevrey regular in overline{Omega }backslash {mathscr {S}} with point singularities concentrated at a set {mathscr {S}}subset overline{Omega } consisting of a finite number of points in overline{Omega }, we prove exponential rates of convergence of hp-version continuous Galerkin finite element methods on affine families of regular, simplicial meshes in Omega . The simplicial meshes are geometrically refined towards {mathscr {S}} but are otherwise unstructured.
Highlights
Many nonlinear PDEs admit solutions which are smooth in a bounded, polytopal domain ⊂ R, but exhibit isolated point singularities at a set S ⊂
Research performed in part while the authors were visiting the Erwin Schrödinger Institute (ESI) in Vienna, Austria, during the ESI thematic term “Numerical Analysis of Complex PDE Models in the Sciences” from June-August 2018
We show that the output of Algorithm 1 can, for each fixed 0 < σ < 1, be identified with sequences Mκ,σ of regular, simplicial meshes which are σ -geometrically refined towards S in
Summary
Many nonlinear PDEs admit solutions which are smooth in a bounded, polytopal domain ⊂ R, but exhibit isolated point singularities at a set S ⊂. We prove an exponential convergence result for C0-conforming hp-FEM on regular, simplicial mesh families with isotropic, geometric refinement towards the singular point(s) c ∈ S. These meshes are in addition required to be shape-regular. The rate (1) coincides, in the cases d = 1, 2 and for analytic solutions, i.e. when δ = 1, with the exponential convergence rate bounds obtained in [18,19] for corner singularities on structured geometric meshes (consisting of axiparallel quadrilaterals with inserted triangles to remove irregular nodes).
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