$hp$-Finite Elements for Fractional Diffusion

Abstract

The purpose of this work is to introduce and analyze a numerical scheme to efficiently solve boundary value problems involving the spectral fractional Laplacian. The approach is based on a reformulation of the problem posed on a semi-infinite cylinder in one more spatial dimension. After a suitable truncation of this cylinder, the resulting problem is discretized with linear finite elements in the original domain and with $hp$-finite elements in the extended direction. The proposed approach yields a drastic reduction of the computational complexity in terms of degrees of freedom and even has slightly improved convergence properties compared to a discretization using linear finite elements for both the original domain and the extended direction. The performance of the method is illustrated by numerical experiments.

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• Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons
• Finite elements for Matérn-type random fields: Uncertainty in computational mechanics and design optimization
• An a posteriori error estimator for the spectral fractional power of the Laplacian
• An exponentially convergent discretization for space-time fractional parabolic equations using $hp$-FEM