Volume 51, Number 6, November-December 2017
|Page(s)||2049 - 2067|
|Published online||23 November 2017|
A note on semilinear fractional elliptic equation: analysis and discretization∗
1 Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA.
2 Chair of Optimal Control, Center of Mathematical Sciences, Technical University of Munich, Boltzmannstraße 3, 85748 Garching by Munich, Germany.
3 University of Puerto Rico (Rio Piedras Campus), College of Natural Sciences, Department of Mathematics, PO Box 70377 San Juan PR 00936-8377 ( USA).
Received: 20 July 2016
Revised: 9 March 2017
Accepted: 20 April 2017
In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order s ∈ (0,1). We identify minimal conditions on the nonlinear term and the source which lead to existence of weak solutions and uniform L∞-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.
Mathematics Subject Classification: 35S15 / 26A33 / 65R20 / 65N12 / 65N30
Key words: Fractional Dirichlet Laplace operator / semi-linear elliptic problems / regularity of weak solutions / discretization / error estimates
© EDP Sciences, SMAI 2017
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