Space-time adaptive finite elements for nonlocal parabolic variational inequalities
This work provides rigorous error analysis and adaptive methods for a class of nonlocal dynamic contact and friction problems, which is a specialized domain within numerical analysis.
The authors develop a priori and a posteriori error estimates for finite element discretizations of nonlocal parabolic variational inequalities involving the fractional Laplacian, and demonstrate that space-time adaptive mesh refinement procedures are efficient for 2D model problems.
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.