Geodesic
A note on semilinear fractional elliptic equation: analysis and discretization
Antil, Harbir1  ; Pfefferer, Johannes2 ; Warma, Mahamadi3 
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).
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2049-2067

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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.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2017023
Classification : 35S15, 26A33, 65R20, 65N12, 65N30
Keywords: Fractional Dirichlet Laplace operator, semi-linear elliptic problems, regularity of weak solutions, discretization, error estimates

Antil, Harbir 1 ; Pfefferer, Johannes 2 ; Warma, Mahamadi 3

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). 
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     title = {A note on semilinear fractional elliptic equation: analysis and discretization},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {2049--2067},
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Antil, Harbir; Pfefferer, Johannes; Warma, Mahamadi. A note on semilinear fractional elliptic equation: analysis and discretization. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2049-2067. doi: 10.1051/m2an/2017023

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