Finite element approximation of the parabolic fractional obstacle problem
Provides a numerical analysis foundation for solving parabolic fractional obstacle problems, which are relevant for modeling nonlocal phenomena in materials science and finance.
The authors develop a finite element approximation for the parabolic fractional obstacle problem, achieving error estimates under various smoothness assumptions by reformulating the fractional Laplacian via a Dirichlet-to-Neumann map on a truncated cylinder.
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