Finite element approximation of the parabolic fractional obstacle problem

TL;DR

This paper introduces a numerical method for approximating the parabolic fractional obstacle problem using finite elements and backward Euler time discretization, with an error analysis based on solution smoothness.

Contribution

It presents a novel discretization approach for the fractional obstacle problem via a truncated nonuniform elliptic formulation and provides an error analysis for the method.

Findings

The method effectively approximates the fractional obstacle problem.

Error estimates depend on solution smoothness assumptions.

The approach handles the nonlocal fractional Laplacian via a local elliptic reformulation.

Abstract

We study a discretization technique for the parabolic fractional obstacle problem in bounded domains. The fractional Laplacian is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic equation posed on a semi-infinite cylinder, which recasts our problem as a quasi-stationary elliptic variational inequality with a dynamic boundary condition. The rapid decay of the solution suggests a truncation that is suitable for numerical approximation. We discretize the truncation with a backward Euler scheme in time and, for space, we use first-degree tensor product finite elements. We present an error analysis based on different smoothness assumptions