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

TL;DR

This paper proves exponential convergence of an $hp$-ILG finite element method for semilinear elliptic boundary value problems with monomial reactions, supported by numerical experiments demonstrating efficiency.

Contribution

It provides a rigorous convergence analysis showing exponential convergence of $hp$-ILG methods for a class of semilinear elliptic problems with corner singularities.

Findings

Exponential convergence in $ ext{H}^1( abla)$ norm for the $hp$-ILG scheme.

Numerical results confirm theoretical exponential convergence rates.

Efficient computational complexity demonstrated through experiments.

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 $\Omega\subset\mathbb{R}^2$ with a finite number of straight edges. In particular, we analyze 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 $\Omega$, with geometric corner refinement, with polynomial degrees increasing in sync with the geometric mesh refinement towards the corners of $\Omega$. 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 $\mathrm{H}^1(\Omega)$ 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.