Space-time adaptive finite elements for nonlocal parabolic variational inequalities

TL;DR

This paper develops and analyzes adaptive finite element methods for nonlocal parabolic variational inequalities involving the fractional Laplacian, providing error estimates and confirming efficiency through numerical experiments.

Contribution

It introduces new a priori and a posteriori error estimates for adaptive schemes applied to nonlocal parabolic inequalities, especially with mixed formulations including contact forces.

Findings

Adaptive methods achieve optimal convergence rates.

A posteriori error estimates effectively guide mesh refinement.

Numerical experiments confirm theoretical error bounds.

Abstract

This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic variational inequalities associated to the fractional Laplacian we obtain a priori and a posteriori error estimates and study the resulting space-time adaptive mesh-refinement procedures. Particular emphasis is placed on mixed formulations, which include the contact forces as a Lagrange multiplier. Corresponding results are presented for elliptic problems. Our numerical experiments for $2$-dimensional model problems confirm the theoretical results: They indicate the efficiency of the a posteriori error estimates and illustrate the convergence properties of space-time adaptive, as well as uniform and graded discretizations.