Exponential convergence in $$H^1$$ of hp-FEM for Gevrey regularity with isotropic singularities

Abstract

AbstractFor functions $$u\in H^1(\Omega )$$u∈H1(Ω) in a bounded polytope $$\Omega \subset {\mathbb {R}}^d$$Ω⊂Rd$$d=1,2,3$$d=1,2,3 with plane sides for $$d=2,3$$d=2,3 which are Gevrey regular in $$\overline{\Omega }\backslash {\mathscr {S}}$$Ω¯\S with point singularities concentrated at a set $${\mathscr {S}}\subset \overline{\Omega }$$S⊂Ω¯ consisting of a finite number of points in $$\overline{\Omega }$$Ω¯, we prove exponential rates of convergence of hp-version continuous Galerkin finite element methods on affine families of regular, simplicial meshes in $$\Omega $$Ω. The simplicial meshes are geometrically refined towards $${\mathscr {S}}$$S but are otherwise unstructured.