Finite element approximations of the nonhomogeneous fractional Dirichlet problem. (3rd May 2018)

• Record Type: Journal Article
• Authors: Acosta, Gabriel; Borthagaray, Juan Pablo; Heuer, Norbert
• Is Part Of: IMA journal of numerical analysis. Volume 39:Number 3(2019)
• Page Start: 1471
• Page End: 1501
• Publication Date: 2018-05-03
• Subjects: fractional Laplacian -- mixed finite elements -- a priori error analysis; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/dry023
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: 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.