2024/04/29 by Yanchen He, He, Yanchen, Paul Houston +5 · 2 citations
Engineering · Mathematics · #65N30 #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2404.18569
openalex publication_date 2024/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 Ω, with geometric corner refinement, with polynomial degrees increasing in sync 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.