A fractional Laplace equation: regularity of solutions and finite element approximations

GABRIEL ACOSTA; JUAN PABLO BORTHAGARAY

Congreso; Wonapde 2016; 2016

In this work we deal with the Dirichlet homogeneous problem for the emph{integral} fractional Laplacian on a bounded domain $Omega subset mathbb{R}^n$. Namely, we deal with basic analytical aspects required to convey a complete Finite Element analysis of the problem egin{equation}leftlbrace egin{array}{l} (-Delta)^s u = f mbox{ in }Omega, \ u = 0 mbox{ in }Omega^c , \ end{array} ight.label{eq:fraccionario}end{equation} where the fractional Laplacian of order $s$ is defined byegin{equation*}(-Delta)^s u (x) = C(n,s) mbox{ P.V.} int_{mathbb{R}^n} rac{u(x)-u(y)}{|x-y|^{n+2s}} dy,end{equation*}and $C(n,s)$ is a normalization constant.Independently of the Sobolev regularity of the source $f$, solutions of eqref{eq:fraccionario} are not expected to be in a better space than $H^{s+min{s,1/2-epsilon}}(Omega)$ (see cite{Grubb, VishikEskin}). However, by building on H"older estimates developed in cite{RosOtonSerra}, we were able to obtain further regularity results in in a novel framework of weighted fractional Sobolev spaces, leading to a priori estimates in terms of the H"older regularity of the data cite{AcostaBorthagaray}. After developing a suitable polynomial interpolation theory in these weighted fractional spaces, optimal order of convergence in the energy norm for the standard linear finite element method is proved for graded meshes. Numerical experiments are in agreement with our theoretical predictions, and illustrate the optimality of the aforementioned estimates.