$hp$-Finite Elements for Fractional Diffusion

Understand

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.