Exponential convergence of hp-ILGFEM for semilinear elliptic boundary value problems with monomial reaction

Abstract

We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $\varOmega \subset{\mathbb{R}}^{2}$ with a finite number of straight edges. In particular, we analyse the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $\varOmega $, with geometric corner refinement, with polynomial degrees increasing in tandem with the geometric mesh refinement towards the corners of $\varOmega $. For a sequence of discrete solutions generated by the ILG solver with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution we prove exponential convergence in $\text{H}^{1}(\varOmega )$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.