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A priori error estimates for finite element discretization of semilinear elliptic equations with non-Lipschitz nonlinearities

Boris Vexler

Abstract

In this paper we develop numerical analysis for finite element discretization of semilinear elliptic equations with potentially non-Lipschitz nonlinearites. The nonlinearity is essecially assumed to be continuous and monotonically decreasing including the case of non-Lipschitz or even non-Hölder continuous nonlinearities. For a direct discretization (without any regularization) with linear finite elements we derive error estimates with respect to various norms and illustrate these results with a numerical example.

A priori error estimates for finite element discretization of semilinear elliptic equations with non-Lipschitz nonlinearities

Abstract

In this paper we develop numerical analysis for finite element discretization of semilinear elliptic equations with potentially non-Lipschitz nonlinearites. The nonlinearity is essecially assumed to be continuous and monotonically decreasing including the case of non-Lipschitz or even non-Hölder continuous nonlinearities. For a direct discretization (without any regularization) with linear finite elements we derive error estimates with respect to various norms and illustrate these results with a numerical example.

Paper Structure

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem