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

Yanchen He, Paul Houston, Christoph Schwab, Thomas P. Wihler

TL;DR

This work develops a fully explicit hp-version iterative linearized Galerkin method for semilinear elliptic boundary value problems with monomial reactions on polygonal domains. By exploiting analytic forcing, corner-weighted Sobolev regularity, and geometric corner refinements, the authors prove exponential convergence in the $H^1$ norm and establish a polylogarithmic $\varepsilon$-complexity bound for the fully discrete scheme. The analysis covers both the nonlinear Galerkin discretization and the linearized solve, including cost reductions via static condensation, and is supported by numerical experiments on square and L-shaped domains that confirm the theoretical rates. The results highlight the practicality of hp-FEM with ILG for fast, accurate solutions of nonlinear elliptic PDEs with analytic data, and they set the stage for hp-adaptive extensions and coefficient-variation generalizations.

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 $Ω\subset\mathbb{R}^2$ 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 $Ω$, 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 $\mathrm{H}^1(Ω)$ 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.

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

TL;DR

This work develops a fully explicit hp-version iterative linearized Galerkin method for semilinear elliptic boundary value problems with monomial reactions on polygonal domains. By exploiting analytic forcing, corner-weighted Sobolev regularity, and geometric corner refinements, the authors prove exponential convergence in the norm and establish a polylogarithmic -complexity bound for the fully discrete scheme. The analysis covers both the nonlinear Galerkin discretization and the linearized solve, including cost reductions via static condensation, and is supported by numerical experiments on square and L-shaped domains that confirm the theoretical rates. The results highlight the practicality of hp-FEM with ILG for fast, accurate solutions of nonlinear elliptic PDEs with analytic data, and they set the stage for hp-adaptive extensions and coefficient-variation generalizations.

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 with a finite number of straight edges. In particular, we analyze the convergence of -type iterative linearized Galerkin (-ILG) solvers. Our convergence analysis is carried out for conforming -finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of , with geometric corner refinement, with polynomial degrees increasing in sync with the geometric mesh refinement towards the corners of . 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 -FE Galerkin solution, we prove exponential convergence in 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.
Paper Structure (18 sections, 11 theorems, 79 equations, 2 figures)

This paper contains 18 sections, 11 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Weak solution
  3. Weak formulation
  4. Well-posedness of weak formulation
  5. Analytic regularity in scales of Hilbertian, corner-weighted Sobolev spaces
  6. Iterative linearized Galerkin (ILG) discretization
  7. Finite-dimensional Galerkin approximations
  8. Iterative solution
  9. Exponential convergence of the ILG iteration
  10. Exponential Convergence and $\varepsilon$-Complexity of $hp$-ILGFEM
  11. $hp$-Approximations on geometric corner meshes
  12. Solution of the nonlinear algebraic system
  13. Evaluation cost of the nonlinear term
  14. Error vs. work for dense Cholesky decomposition
  15. Static condensation Cholesky decomposition
  16. ...and 3 more sections

Key Result

Lemma 1

For any $q\in\mathbb{N}_0$ the nonlinear form from eq:b satisfies the monotonicity property Furthermore, the (local) Lipschitz continuity bound holds for any $u,v,w\in\mathrm{H}^1_{\mathcal{D}}(\Omega)$, with $C_q(\Omega)$ the constant from eq:Sobolev.

Figures (2)

  • Figure 1: Example 1 (square domain): (a) Comparison of the error with respect to the third root of the number of degrees of freedom; (b) Comparison of the error with respect to the seventh root of the work (CPU time) measured in seconds.
  • Figure 2: Example 2 (L-shaped domain): (a) Comparison of the error with respect to the third root of the number of degrees of freedom; (b) Comparison of the error with respect to the seventh root of the work (CPU time) measured in seconds.

Theorems & Definitions (29)

  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1: Existence, uniqueness, and stability
  • proof
  • Definition 1: Weighted analytic class $\mathrm{B}^\ell_{\bm{\mathsf{{\beta}}}}(\Omega)$
  • Proposition 2: Regularity of weak solution
  • Remark 2
  • Proposition 3: Well-posedness of Galerkin discretization
  • Proposition 4: Quasi-optimality
  • ...and 19 more