A note on semilinear fractional elliptic equation: analysis and discretization

TL;DR

This paper investigates the existence, regularity, and numerical approximation of solutions to fractional semilinear elliptic equations, providing theoretical insights and finite element methods with error estimates.

Contribution

It establishes minimal conditions for solution existence and bounds, and introduces a tensor product finite element approach with error analysis for fractional elliptic equations.

Findings

Existence and boundedness of weak solutions under minimal conditions

Development of a finite element discretization with error estimates

Numerical example demonstrating the method's effectiveness

Abstract

In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order $s \in (0,1)$. We identify minimal conditions on the nonlinear term and the source which leads to existence of weak solutions and uniform $L^\infty$-bound on the solutions. Next we realize the fractional Laplacian as a Dirichlet-to-Neumann map via the Caffarelli-Silvestre extension. We introduce a first-degree tensor product finite elements space to approximate the truncated problem. We derive a priori error estimates and conclude with an illustrative numerical example.