A reduced basis method for fractional diffusion operators II

Tobias Danczul; Joachim Schöberl

doi:10.1515/jnma-2020-0042

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A reduced basis method for fractional diffusion operators II

Tobias Danczul und Joachim Schöberl

Veröffentlicht/Copyright: 2. Dezember 2021

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Aus der Zeitschrift Journal of Numerical Mathematics Band 29 Heft 4

Abstract

We present a novel numerical scheme to approximate the solution map s ↦ u(s) := 𝓛^–sf to fractional PDEs involving elliptic operators. Reinterpreting 𝓛^–s as an interpolation operator allows us to write u(s) as an integral including solutions to a parametrized family of local PDEs. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. The integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation L of the operator whose inverse is projected to the s-independent reduced space, where explicit diagonalization is feasible. Exponential convergence rates are proven rigorously.

A second algorithm is presented to avoid inversion of L. Instead, we directly project the matrix to the subspace, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.

Keywords: fractional Laplace; reduced basis method; interpolation spaces; Zolotarëv points; finite element method

MSC 2010: 46B70; 65N12; 65N15; 65N30; 35J15; 65N25; 35P10

Acknowledgment

The authors acknowledge support from the Austrian Science Fund (FWF) through grant No. F65 and W1245.

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6 Appendix

Proof of Theorem 2.1

It suffices to show that

K(V0,V1)2(t;u)=∥u∥02−〈u,v(t)〉0. (6.1)

There holds

∥u−v(t)∥02=∥u∥02−2〈u,v(t)〉0+∥v(t)∥02.

Let u_k := 〈u, φ_k〉₀ to deduce from Lemma 2.1

t2∥v(t)∥12=∑k=1∞t2λk2uk2(1+t2λk2)2=∑k=1∞uk21+t2λk2−uk2(1+t2λk2)2=〈u,v(t)〉0−∥v(t)∥02

which proves (6.1) and concludes the proof.□

Proof of Theorem 2.2

One observes that for any F ∈ 𝒱₀ we have

〈F,φk〉=〈R1F,φk〉1=λk2〈R1F,φk〉0=λk2〈Φk,R1F〉=λk2〈Φk,F〉−1

from which we conclude that (λkΦk)k=1∞ is a 𝒱_–1-orthonormal system of eigenfunctions. Since

0=〈F,Φk〉−1=λk−2〈F,φk〉

for all k ∈ ℕ implies that F = 0, it is also a basis. This proves the claim.□

Proof of Theorem 2.3

Due to

〈F,Φk〉−1=λk−2〈F,φk〉

and Theorem 2.2, there holds

∥F∥H1−s(V−1,V0)2=∑k=1∞λk2−2s〈F,Ψk〉−12=∑k=1∞λk−2s〈F,φk〉2=∥F∥H−s(V0,V1)2

proving the first equality in (2.7). The second one follows by means of (2.2). Furthermore, one observes

R1LH1−s(V−1,V0)F=R1∑k=1∞λk2−2s〈F,Ψk〉−1Ψk=R1∑k=1∞λk−2s〈F,ψk〉Ψk=∑k=1∞λk2−2s〈F,φk〉R1Φk=∑k=1∞λk−2s〈F,φk〉φk=LH−s(V0,V1)F

confirming the first equality in (2.8). The latter is a consequence of (2.3).

The remainder follows as LH−s(V0,V1)F=LHs(V0,V1)−1f for F ∈ 𝒱₀.□

Received: 2020-06-10

Revised: 2021-01-19

Accepted: 2021-02-23

Published Online: 2021-12-02

Published in Print: 2021-10-20

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https://doi.org/10.1515/jnma-2020-0042

Schlagwörter für diesen Artikel

fractional Laplace; reduced basis method; interpolation spaces; Zolotarëv points; finite element method