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Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains. (English) Zbl 1356.65239

The authors consider the numerical approximation of solutions of fractional Laplacian equations on bounded domains. Such equations allow global interactions between points separated by arbitrarily large distances. Two approximations are introduced. First, interactions are localized so that only points less than some specified distance, referred to as the interaction radius, are allowed to interact. The resulting truncated problem is a special case of a more general nonlocal diffusion problem. The second approximation is the spatial discretization of the related nonlocal diffusion problem. Using a recently developed abstract framework for asymptotically compatible schemes, some convergence results for solutions of the truncated and discretized problem to the solutions of the fractional Laplacian problems are proved. It is shown that conforming Galerkin finite element approximations of the nonlocal diffusion equation are always asymptotically compatible schemes for the corresponding fractional Laplacian model as the interaction radius increases and the grid size decreases. The results are obtained under minimal regularity assumptions on the solution which are applicable to general domains and general geometric meshes with no restriction on the space dimension and with data that are only assumed to be square integrable. In addition, the obtained results solve an open conjecture given in the literature about the convergence of numerical solutions on a fixed mesh as the interaction radius increases.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35R11 Fractional partial differential equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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