Finite element approximations of the nonhomogeneous fractional Dirichlet problem
Gabriel Acosta, Juan Pablo Borthagaray, Norbert Heuer · IMA Journal of Numerical Analysis · 2018
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 analogue 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 for 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.