Juan Pablo Borthagaray

A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations

This paper deals with the integral version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the Holder regularity of the data. By relying on these results, optimal order of convergence for the standard linear finite element method is proved for quasi-uniform as well as graded meshes. Some numerical examples are given showing results in agreement with the theoretical predictions.

A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian

In Acosta etal. (2017), a complete n-dimensional finite element analysis of the homogeneous Dirichlet problem associated to a fractional Laplacian was presented. Here we provide a comprehensive and simple 2D MATLAB finite element code for such a problem. The code is accompanied with a basic discussion of the theory relevant in the context. The main program is written in about 80 lines and can be easily modified to deal with other kernels as well as with time dependent problems. The present work fills a gap by providing an input for a large number of mathematicians and scientists interested in numerical approximations of solutions of a large variety of problems involving nonlocal phenomena in two-dimensional space.

Finite Element Approximation for the Fractional Eigenvalue Problem

The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order of such convergence. Finally, we perform some numerical experiments and compare our results with previous work by other authors.

Numerical Methods for Fractional Diffusion

We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford–Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.

Finite element approximations of the nonhomogeneous fractional Dirichlet problem

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.