The pollution effect for FEM approximations of the Ginzburg-Landau equation

TL;DR

This paper analyzes how finite element approximations of the Ginzburg-Landau equation are affected by a pollution effect, showing that higher polynomial degrees can mitigate this issue and providing explicit conditions for accurate solutions.

Contribution

It offers a new error analysis that quantifies the pollution effect and pre-asymptotic regime, demonstrating how polynomial degree influences approximation accuracy.

Findings

Higher polynomial degrees reduce the pollution effect.

Explicit resolution conditions for accurate finite element solutions.

Numerical examples confirm theoretical error estimates.

Abstract

In this paper, we investigate the approximation properties of solutions to the Ginzburg-Landau equation (GLE) in finite element spaces. Special attention is given to how the errors are influenced by coupling the mesh size $h$ and the polynomial degree $p$ of the finite element space to the size of the so-called Ginzburg-Landau material parameter $\kappa$. As observed in previous works, the finite element approximations to the GLE are suffering from a numerical pollution effect, that is, the best-approximation error in the finite element space converges under mild coupling conditions between $h$ and $\kappa$, whereas the actual finite element solutions possess poor accuracy in a large pre-asymptotic regime which depends on $\kappa$. In this paper, we provide a new error analysis that allows us to quantify the pre-asymptotic regime and the corresponding pollution effect in terms of explicit resolution conditions. In particular, we are able to prove that higher polynomial degrees reduce the pollution effect, i.e., the accuracy of the best-approximation is achieved under relaxed conditions for the mesh size. We provide both error estimates in the $H^1$- and the $L^2$-norm and we illustrate our findings with numerical examples.