A numerical energy minimisation approach for semilinear diffusion-reaction boundary value problems based on steady state iterations

TL;DR

This paper introduces an energy minimisation-based numerical method for semilinear diffusion-reaction boundary value problems, employing adaptive mesh refinement driven by energy considerations, and proves convergence with demonstrated robustness.

Contribution

It develops a novel energy-based iterative scheme with adaptive mesh refinement for semilinear problems, avoiding a posteriori error indicators and ensuring convergence.

Findings

Method converges to a local energy minimum.

Adaptive refinement improves solution accuracy.

Numerical experiments confirm robustness and reliability.

Abstract

We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More specifically, this procedure aims to generate a sequence of numerical approximations, which results from the iterative solution of related (stabilised) linearised discrete problems, and tends to a local minimum of the underlying energy functional. Simultaneously, the finite-dimensional approximation spaces are adaptively refined; this is implemented in terms of a new mesh refinement strategy in the context of finite element discretisations, which again relies on the energy structure of the problem under consideration, and does not involve any a posteriori error indicators. In combination, the resulting adaptive algorithm consists of an iterative linearisation procedure on a sequence of hierarchically refined discrete spaces, which we prove to converge towards a solution of the continuous problem in an appropriate sense. Numerical experiments demonstrate the robustness and reliability of our approach for a series of examples.