242 citations · 611 across the 19 of their papers we have counts for
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Finite element approximation for the Bessel -Laplacian
Juan Pablo Borthagaray, José Camilo Rueda Niño
We study the finite element approximation of the Dirichlet problem for the Bessel -Laplacian, , built f…
A mixed formulation for the fractional Poisson problem
Juan Pablo Borthagaray, Nahuel de León
The mixed formulation of the classical Poisson problem introduces the flux as an additional variable, leading to a system of coupled equations. Using fractional calculus identities…
A deep first-order system least squares method for the obstacle problem
Gabriel Acosta, Eugenia Belén, Francisco M. Bersetche +1
We propose a deep learning approach to the obstacle problem inspired by the first-order system least-squares (FOSLS) framework. This method reformulates the problem as a convex min…
On some coupled local and nonlocal diffusion models
Juan Pablo Borthagaray, Patrick Ciarlet
We study problems in which a local model is coupled with a nonlocal one. We propose two energies: both of them are based on the same classical weighted -semi norm to model the…
Monotone two-scale methods for a class of integrodifferential operators and applications
Juan Pablo Borthagaray, Ricardo H. Nochetto, Abner J. Salgado +1
We develop a monotone, two-scale discretization for a class of integrodifferential operators of order , . We apply it to develop numerical schemes, and derive poin…
Fractional Elliptic Problems on Lipschitz Domains: Regularity and Approximation
Juan Pablo Borthagaray, Wenbo Li, Ricardo H. Nochetto
This survey hinges on the interplay between regularity and approximation for linear and quasi-linear fractional elliptic problems on Lipschitz domains. For the linear Dirichlet int…
A deep first-order system least squares method for solving elliptic PDEs
Francisco M. Bersetche, Juan Pablo Borthagaray
We propose a First-Order System Least Squares (FOSLS) method based on deep-learning for numerically solving second-order elliptic PDEs. The method we propose is capable of dealing…
Constructive approximation on graded meshes for the integral fractional Laplacian
Juan Pablo Borthagaray, Ricardo H. Nochetto
We consider the homogeneous Dirichlet problem for the integral fractional Laplacian . We prove optimal Sobolev regularity estimates in Lipschitz domains provided the soluti…
Finite element algorithms for nonlocal minimal graphs
Juan Pablo Borthagaray, Wenbo Li, Ricardo H. Nochetto
We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a…
Robust BPX Preconditioner for Fractional Laplacians on Bounded Lipschitz Domains
Juan Pablo Borthagaray, Ricardo H. Nochetto, Shuonan Wu +1
We propose and analyze a robust BPX preconditioner for the integral fractional Laplacian on bounded Lipschitz domains. For either quasi-uniform grids or graded bisection grids, we…
Finite element approximation of fractional Neumann problems
Francisco M. Bersetche, Juan Pablo Borthagaray
In this paper we consider approximations of Neumann problems for the integral fractional Laplacian by continuous, piecewise linear finite elements. We analyze the weak formulation…
Local energy estimates for the fractional Laplacian
Juan Pablo Borthagaray, Dmitriy Leykekhman, Ricardo H. Nochetto
The integral fractional Laplacian of order is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic…
The -tensor Model with Uniaxial Constraint
Juan Pablo Borthagaray, Shawn W. Walker
This chapter is about the modeling of nematic liquid crystals (LCs) and their numerical simulation. We begin with an overview of the basic physics of LCs and discuss some of their…
A structure-preserving FEM for the uniaxially constrained -tensor model of nematic liquid crystals
Juan Pablo Borthagaray, Ricardo H. Nochetto, Shawn W. Walker
We consider the one-constant Landau - de Gennes model for nematic liquid crystals. The order parameter is a traceless tensor field , which is constrained to be uniaxial…
Linear and Nonlinear Fractional Elliptic Problems
Juan Pablo Borthagaray, Wenbo Li, Ricardo H. Nochetto
This paper surveys recent analytical and numerical research on linear problems for the integral fractional Laplacian, fractional obstacle problems, and fractional minimal graphs. T…
Finite element discretizations of nonlocal minimal graphs: convergence
Juan Pablo Borthagaray, Wenbo Li, Ricardo H. Nochetto
In this paper, we propose and analyze a finite element discretization for the computation of fractional minimal graphs of order~ on a bounded domain . Such a Plat…
On the convergence in -norm for the fractional Laplacian
Juan Pablo Borthagaray, Patrick Ciarlet
We consider the numerical solution of the fractional Laplacian of index in a bounded domain with homogeneous boundary conditions. Its solution a priori belongs to…
Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian
Juan Pablo Borthagaray, Ricardo H. Nochetto, Abner J. Salgado
We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian. The weight is a power of the distance to the…
Finite element approximations of the nonhomogeneous fractional Dirichlet problem
Gabriel Acosta, Juan Pablo Borthagaray, Norbert Heuer
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…
Numerical Methods for Fractional Diffusion
Andrea Bonito, Juan Pablo Borthagaray, Ricardo H. Nochetto +2
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 approa…
Finite element approximations for fractional evolution problems
Gabriel Acosta, Francisco M. Bersetche, Juan Pablo Borthagaray
This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided…
A short FE implementation for a 2d homogeneous Dirichlet problem of a Fractional Laplacian
Gabriel Acosta, Francisco M. Bersetche, Juan Pablo Borthagaray
In \cite{AcostaBorthagaray}, a complete -dimensional finite element analysis of the homogeneous Dirichlet problem associated to a fractional Laplacian was presented. Here we pro…
Regularity theory and high order numerical methods for the (1d)-Fractional Laplacian
Gabriel Acosta, Juan Pablo Borthagaray, Oscar Bruno +1
This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization…
Finite element approximation for the fractional eigenvalue problem
Juan Pablo Borthagaray, Leandro M. Del Pezzo, Sandra Martínez
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 pro…