Numerical approximation of fractional powers of elliptic operators

Abstract

We present and study a novel numerical algorithm to approximate the action of T β := L −β where L is a symmetric and positive definite unbounded operator on a Hilbert space H 0 . The numerical method is based on a representation formula for T −β in terms of Bochner integrals involving (I + t 2 L) −1 for t ∈ (0, ∞).To develop an approximation to T β , we introduce a finite element approximation L h to L and base our approximation to T β on T β h := L −β h . The direct evaluation of T β h is extremely expensive as it involves expansion in the basis of eigenfunctions for L h . The above mentioned representation formula holds for T −β h and we propose three quadrature approximations denoted generically by Q β h . The two results of this paper bound the errors in the H 0 inner product of T β − T β h π h and T β h − Q β h where π h is the H 0 orthogonal projection into the finite element space. We note that the evaluation of Q β h involves application of (I + (t i ) 2 L h ) −1 with t i being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. Numerical experiments illustrating the theoretical estimates are provided for both the quadrature error T β h − Q β h and the finite element error T β − T β h π h .

Optimal solvers for linear systems with fractional powers of sparse SPD matrices

“…Such error bounds are derived under certain assumptions that are an interplay among data regularity, regularity pickup of , and fractional power α . The needed justification is provided by the work of Bonito et al, for the problem in the case when q =0 and Γ D = ∂ Ω; see also an earlier work . Following the work of Bonito et al, one introduces and shows that is a Hilbert space under the inner product , for all .…”

“…The needed justification is provided by the work of Bonito et al, for the problem in the case when q =0 and Γ D = ∂ Ω; see also an earlier work . Following the work of Bonito et al, one introduces and shows that is a Hilbert space under the inner product , for all . To set up a finite element approximation of , we first introduce its weak form: find such that …”

“…Here, for , we use an expression similar to but involving the eigenfunctions and eigenvalues of . As shown in the work of Bonito et al, if the operator satisfies the regularity pickup, that is, there is s ∈(0,1] such that , and is a bounded map of into , then for α > s , one has …”

“…The paper (see theorem 4.3 in the work of Bonito et al) contains more refined results depending on the relationship among smoothness of the data δ , the regularity pickup s , and the fractional order α . In the case of full regularity, s =1, the best possible rate for f ∈ L 2 (Ω) is (cf.…”

A finite difference method for fractional diffusion equations with Neumann boundary conditions

“…4. An alternative approximation of fractional order elliptic operators (which can be applied in multidimensional cases) was proposed in [34], which can be a basis also for finite element discretizations.…”