Quasi-Optimality of an Adaptive Finite Element Method for Cathodic Protection
This work provides a theoretical guarantee of quasi-optimal convergence for adaptive finite element methods in a specific engineering application (cathodic protection), which is an incremental contribution to the field of adaptive finite element analysis.
The authors derived a reliable and efficient error estimator for adaptive finite element methods applied to a 2D cathodic protection problem with nonlinear boundary conditions, and proved quasi-optimal convergence rates, which were confirmed by numerical experiments.
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 Dörfler 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.