Adaptive Pseudo-Transient-Continuation-Galerkin Methods for Semilinear Elliptic Partial Differential Equations

TL;DR

This paper introduces an adaptive Pseudo-Transient-Continuation-Galerkin method for solving semilinear elliptic PDEs, combining residual analysis, adaptive discretization, and numerical validation to enhance robustness and efficiency.

Contribution

It presents a novel fully adaptive PTC-Galerkin scheme integrating residual reduction analysis and adaptive finite element discretization for semilinear elliptic equations.

Findings

Robustness demonstrated across various examples

Effective residual-based adaptive discretization

Reliable convergence in numerical experiments

Abstract

In this paper we investigate the application of pseudo-transient-continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, employ the PTC-methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction-type PTC-method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.