On the convergence in $H^1$-norm for the fractional Laplacian
Provides theoretical justification for H^1-norm error estimates in fractional Laplacian finite element methods, which is an incremental advance for numerical analysts working on fractional PDEs.
The paper derives error estimates in the H^1-norm for Lagrange finite element solutions of the fractional Laplacian, showing optimal convergence rates on quasi-uniform and graded meshes.
We consider the numerical solution of the fractional Laplacian of index $s\in(1/2,1)$ in a bounded domain $Ω$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space ${\widetilde H}^s(Ω)$. For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in $H^1(Ω)$. 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 $H^1(Ω)$. A natural question is then whether one can obtain error estimates in $H^1(Ω)$-norm, in addition to the classical ones that can be derived in the ${\widetilde H}^s(Ω)$ 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.