Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

Abstract

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment.

Keywords

• fractional Laplacian
• elliptic boundary value problems
• boundary integral equations
• boundary element methods
• Riesz potential