Functional a Posteriori Estimates for the Fractional Laplacian Problem

Abstract The paper is concerned with a posteriori estimates for approximations of boundary value problems generated by the spectral fractional Laplace operator. The derivation is based upon the Stinga–Torrea extension, which generalizes the Caffarelli–Silvestre extension and transfers the corresponding nonlocal problem in a bounded domain to a local problem of higher dimensionality. A posteriori estimates are first derived for this local problem. Two–sided error bounds for the original problem follow from them. The estimates are fully computable and contain no conditions and constants depending on a method or mesh used to compute an approximation. They are valid for any energy admissible approximation of the extended problem.