A reduced basis method for fractional diffusion operators

Abstract

in Englisch

Several authors have proposed and analyzed numerical methods for fractional differential oper- ators, in particular Fourier Galerkin schemes and Caffarelli-Silvestre extensions. In this thesis we consider a different approach. By means of a reduced basis method, the desired operator is projected to a low dimensional space V r , where the fractional power can be directly evaluated via the eigen-system. The optimal choice of V r is provided by the so called Zolotarëv points, en- suring exponential convergence. Numerical experiments evaluating the operator and the inverse operator confirm the analysis. The time-dependent Fractional Cahn-Hilliard Equation (FCHE) is examined for further tests. By a splitting method, the non-linear operator is decoupled from the regular Laplacian, such that the linear parabolic equation is solved exactly on the low dimensional reduced space. Different choices of the fractional power s are discussed and tested.