Quasi-Optimality of an Adaptive Finite Element Method for Cathodic Protection

TL;DR

This paper develops a reliable adaptive finite element method for a 2D cathodic protection problem, demonstrating quasi-optimal convergence rates through theoretical analysis and numerical experiments.

Contribution

It introduces a new residual-based error estimator and proves the quasi-optimality of the adaptive algorithm for nonlinear boundary problems.

Findings

The adaptive method achieves quasi-optimal convergence rates.

Numerical experiments confirm the theoretical results.

The approach effectively handles nonlinear boundary conditions.

Abstract

In this work, we derive a reliable and efficient residual-typed error estimator for the finite element approximation of a 2d cathodic protection problem governed by a steady-state diffusion equation with a nonlinear boundary condition. We propose a standard adaptive finite element method involving the D\"{o}rfler marking and a minimal refinement without the interior node property. Furthermore, we establish the contraction property of this adaptive algorithm in terms of the sum of the energy error and the scaled estimator. This essentially allows for a quasi-optimal convergence rate in terms of the number of elements over the underlying triangulation. Numerical experiments are provided to confirm this quasi-optimality.