FE approximations of the nonhomogeneous fractional Dirichlet problem

N. HEUER; G. ACOSTA; J. P. BORTHAGARAY

Santiago de Chile

Congreso; Santiago Numérico III; 2017

Pontificia Universidad Católica, Santiago de Chile

We study finite element approximations of the following non-homogeneous Dirichlet problem(−∆)su = f in Ω,u = g in Ωcon a bounded domain Ω ⊂ Rn. The operator (−∆)s stands for the Fractional Laplacianand the functions f and g belong to suitable spaces. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of thenormal derivative as a Lagrange multiplier in the formulation of the problem. In orderto obtain convergence orders for our scheme, regularity estimates are developed, bothfor the solution and its nonlocal derivative. The method we propose requires that, asmeshes are refined, the discrete problems be solved in a family of domains of growing diameter.