Exponential convergence of hp-ILGFEM for semilinear elliptic boundary value problems with monomial reaction
Publisher DOI
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 Ω ⊂ R2 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 Ω, with geometric corner refinement, with polynomial degrees increasing in tandem with the geometric mesh refinement
towards the corners of Ω. 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 H1(Ω) 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.
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 Ω, with geometric corner refinement, with polynomial degrees increasing in tandem with the geometric mesh refinement
towards the corners of Ω. 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 H1(Ω) 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.
Date Issued
2025-06-26
Publication Type
Article
Subject(s)
Subjects
semilinear elliptic PDE in polygons
•
monomial reactions
•
monotone problems
•
hp-iterative linearized Galerkin methods
•
exponential convergence
Language(s)
en
Author(s)
He, Yanchen | |
Houston, Paul | |
Schwab, Christoph |
Journal
IMA Journal of Numerical Analysis
Publisher
Oxford University Press
ISSN
0272-4979
1464-3642
Access(Rights)
open.access