A posteriori error estimates for parabolic partial differential equations on stationary surfaces

TL;DR

This paper introduces a residual-based a posteriori error estimator for parabolic PDEs on stationary surfaces, enabling adaptive algorithms with proven error bounds and validated by numerical experiments.

Contribution

It develops a novel error estimator for surface parabolic PDEs and proposes an adaptive algorithm with theoretical error bounds and numerical validation.

Findings

Error estimator bounds the error globally in space and time.

Adaptive algorithm based on the estimator improves solution accuracy.

Numerical experiments confirm the effectiveness of the proposed method.

Abstract

This paper develops and discusses a residual-based a posteriori error estimator for parabolic surface partial differential equations on closed stationary surfaces. The full discretization uses the surface finite element method in space and the backward Euler method in time. The proposed error indicator bounds the error quantities globally in space from above and below, and globally in time from above and locally from below. Based on the derived error indicator, a space-time adaptive algorithm is proposed. Numerical experiments illustrate and complement the theory.