Fractional Elliptic Quasi-Variational Inequalities: Theory and Numerics

TL;DR

This paper develops a new theoretical framework and numerical methods for fractional elliptic quasi-variational inequalities, addressing existence, uniqueness, and solution approximation via a Dirichlet-to-Neumann map approach.

Contribution

It introduces a novel paradigm for fractional QVIs by extending them to a semi-infinite cylinder and analyzing convergence of truncated solutions.

Findings

Existence and uniqueness of solutions for the extended fractional QVI.

Convergence of truncated solutions to the original problem as truncation parameter increases.

Development of a convergent numerical algorithm for solving the truncated problem.

Abstract

This paper introduces an elliptic quasi-variational inequality (QVI) problem class with fractional diffusion of order $s \in (0,1)$, studies existence and uniqueness of solutions and develops a solution algorithm. As the fractional diffusion prohibits the use of standard tools to approximate the QVI, instead we realize it as a Dirichlet-to-Neumann map for a problem posed on a semi-infinite cylinder. We first study existence and uniqueness of solutions for this extended QVI and then transfer the results to the fractional QVI: This introduces a new paradigm in the field of fractional QVIs. Further, we truncate the semi-infinite cylinder and show that the solution to the truncated problem converges to the solution of the extended problem, under fairly mild assumptions, as the truncation parameter $\tau$ tends to infinity. Since the constraint set changes with the solution, we develop an argument using Mosco convergence. We state an algorithm to solve the truncated problem and show its convergence in function space. Finally, we conclude with several illustrative numerical examples.