Finite element approximations of the nonhomogeneous fractional Dirichlet problem
This work provides a numerical scheme for a class of fractional PDEs, but the results are incremental and domain-specific to fractional Laplacian problems.
The authors develop finite element approximations for the nonhomogeneous fractional Dirichlet problem using a weak imposition of Dirichlet conditions and a Lagrange multiplier for the nonlocal normal derivative. They derive regularity estimates and achieve convergence orders, with the method requiring solving discrete problems on domains of increasing diameter as meshes are refined.
We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative as a Lagrange multiplier in the formulation of the problem. In order to obtain convergence orders for our scheme, regularity estimates are developed, both for the solution and its nonlocal derivative. The method we propose requires that, as meshes are refined, the discrete problems be solved in a family of domains of growing diameter.