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On the convergence in $H^1$-norm for the fractional Laplacian

Juan Pablo Borthagaray 1, 2 Patrick Ciarlet 3
3 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : We consider the numerical solution of the fractional Laplacian of index $s \in (1/2, 1)$ in a bounded domain $\Omega$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space $\widetilde{H}^s(\Omega)$. For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in $H^1(\Omega)$. In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to $H^1(\Omega)$. A natural question is then whether one can obtain error estimates in $H^1(\Omega)$-norm, in addition to the classical ones that can be derived in the $\widetilde{H}^s(\Omega)$ energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes.
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Juan Pablo Borthagaray, Patrick Ciarlet. On the convergence in $H^1$-norm for the fractional Laplacian. SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2019, 57, pp.1723-1743. ⟨hal-01912092⟩

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