A note on semilinear fractional elliptic equation: analysis and discretization

Antil, Harbir; Pfefferer, Johannes; Warma, Mahamadi

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2049-2067.

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 lead 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.

Classification : 35S15, 26A33, 65R20, 65N12, 65N30
Mots clés : Fractional Dirichlet Laplace operator, semi-linear elliptic problems, regularity of weak solutions, discretization, error estimates

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