# Abstract

We present a novel numerical scheme to approximate the solution map *s* ↦ u(*s*) := 𝓛^{−s}*f* 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.

# Acknowledgements

The authors acknowledge support from the Austrian Science Fund (FWF) through grant number F 65 and W1245.

### References

[1] M. Abramowitz and I. A. Stegun, *Handbook of mathematical functions with formulas, graphs, and mathematical tables*, vol. 55, National Bureau of Standards Applied Mathematics Series, 1964. Search in Google Scholar

[2] M. Ainsworth and C. Glusa, *Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver*, Computer Methods in Applied Mechanics and Engineering (2017). Search in Google Scholar

[3] M. Ainsworth and C. Glusa, *Hybrid finite element* - *spectral method for the fractional Laplacian: Approximation theory and efficient solver*, SIAM Journal on Scientific Computing **40** (2017). Search in Google Scholar

[4] M. Ainsworth and C. Glusa, *Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains*, pp. 17–57, Springer International Publishing, 2018. Search in Google Scholar

[5] V. Anh, M. Ilic, F. Liu, and I. Turner, *Numerical approximation of a fractional*-*in*-*space diffusion equation (II)* - *with nonhomogeneous boundary conditions*, Fractional Calculus and Applied Analysis **9** (2006). Search in Google Scholar

[6] H. Antil and S. Bartels, *Spectral approximation of fractional PDEs in image processing and phase field modeling*, Comput. Methods Appl. Math. (2017). Search in Google Scholar

[7] H. Antil, Y. Chen, and A. C. Narayan, *Reduced basis methods for fractional Laplace equations via extension*, SIAM J. Scientific Computing **41** (2018), A3552–A3575. Search in Google Scholar

[8] H. Antil and C. Rautenberg, *Sobolev spaces with non-muckenhoupt weights, fractional elliptic operators, and applications*, SIAM Journal on Mathematical Analysis **51** (2018). Search in Google Scholar

[9] D. Applebaum, *Lévy processes*—*from probability to finance and quantum groups*, Notices Amer. Math. Soc. **51** (2004), no. 11, 1336–1347. Search in Google Scholar

[10] L. Banjai, J. M. Melenk, R. H. Nochetto, E. Otárola, A. J. Salgado, and C. Schwab, *Tensor FEM for spectral fractional diffusion*, Found. of Comput. Math. (2018). Search in Google Scholar

[11] P. W. Bates, *On some nonlocal evolution equations arising in materials science*, Nonlinear dynamics and evolution equations (2006), 13–52. Search in Google Scholar

[12] D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, *Application of a fractional advection*-*dispersion equation*, Soil Science Society of America Journal **36** (2000). Search in Google Scholar

[13] H. Berestycki, J. Roquejoffre, and L. Rossi, *The periodic patch model for population dynamics with fractional diffusion*, Discrete and Continuous Dynamical Systems - S **4** (2011). Search in Google Scholar

[14] J. Bergh and J. Lofstrom, *Interpolation spaces*, Springer-Verlag, Berlin, 1976. Search in Google Scholar

[15] A. Bonito, J. Borthagaray, R. H. Nochetto, E. Otarola, and A. J. Salgado, *Numerical methods for fractional diffusion*, Computing and Visualization in Science (2017). Search in Google Scholar

[16] A. Bonito, D. Guignard, and A. R. Zhang, *Reduced basis approximations of the solutions to fractional diffusion problems*, arXiv:1905.01754 [math.NA] (2019). Search in Google Scholar

[17] A. Bonito, W. Lei, and J. E. Pasciak, *On sinc quadrature approximations of fractional powers of regularly accretive operators*, Journal of Numerical Mathematics (2018). Search in Google Scholar

[18] A. Bonito and J. E. Pasciak, *Numerical approximation of fractional powers of elliptic operators*, Math. Comput. **84** (2015), 2083–2110. Search in Google Scholar

[19] J. H. Bramble, *Multigrid methods*, Pitman research notes in mathematics, New York, 1993. Search in Google Scholar

[20] C. Brändle, E. Colorado, A. de Pablo, and U. Sánchez, *A concave*-*convex elliptic problem involving the fractional Laplacian*, Proceedings of the Royal Society of Edinburgh Section A Mathematics **43** (2010). Search in Google Scholar

[21] A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez, and K. Burrage, *Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization*, J. R. Soc. Interface (2014). Search in Google Scholar

[22] X. Cabré and J. Tan, *Positive solutions of nonlinear problems involving the square root of the Laplacian*, Advances in Mathematics **224** (2010). Search in Google Scholar

[23] L. Caffarelli and L. Silvestre, *An extension problem related to the fractional Laplacian*, Communications in Partial Differential Equations **32** (2007), no. 8, 1245–1260. Search in Google Scholar

[24] A. Capella, J. Dávila, L. Dupaigne, and Y. Sire, *Regularity of radial extremal solutions for some non*-*local semilinear equations*, Communications in Partial Differential Equations **36** (2011), no. 8, 1353–1384. Search in Google Scholar

[25] L. Chen, R. H. Nochetto, E. Otarola, and A. J. Salgado, *Multilevel methods for nonuniformly elliptic operators and fractional diffusion*, Mathematics of Computation **85** (2016). Search in Google Scholar

[26] T. Danczul and J. Schöberl, *A reduced basis method for fractional diffusion operators I*, arXiv:1904.05599v2 [math.NA] (2019). Search in Google Scholar

[27] H. Dinh, H. Antil, Y. Chen, E. Cherkaev, and A. Narayan, *Model reduction for fractional elliptic problems using Kato’s formula*, arXiv:1904.09332 [math.NA] (2019). Search in Google Scholar

[28] M. E. Farquhar, T. J. Moroney, Q. Yang, I. W. Turner, and K. F. Burrage, *Computational modelling of cardiac ischaemia using a variable*-*order fractional Laplacian*, arXiv:1809.07936v1 [math.NA] (2018). Search in Google Scholar

[29] M. Faustmann, J. M. Melenk, and D. Praetorius, 𝓗-*matrix approximability of the inverses of FEM matrices*, Numerische Mathematik **131** (2015), 615–642. Search in Google Scholar

[30] M. Faustmann, J. M. Melenk, and D. Praetorius, *Quasi*-*optimal convergence rate for an adaptive method for the integral fractional Laplacian*, arXiv:1903.10409 [math.NA] (2019). Search in Google Scholar

[31] G. Gilboa and S. Osher, *Nonlocal operators with applications to image processing*, Multiscale Modeling and Simulation **7** (2009), no. 3, 1005–1028. Search in Google Scholar

[32] A. A. Gonchar, *Zolotarëv problems connected with rational functions*, Mathematics of the USSR-Sbornik **78** (120) (1969), 640–654. Search in Google Scholar

[33] J. S. Hesthaven, G. Rozza, and B. Stamm, *Certified reduced basis methods for parametrized partial differential equations*, 1 ed., Springer, Switzerland, 2015. Search in Google Scholar

[34] C. Hofreither, *A unified view of some numerical methods for fractional diffusion*, Comput. Math. Appl. (2019). Search in Google Scholar

[35] M. Ilić, F. Liu, I. Turner, and V. Anh, *Numerical approximation of a fractional*-*in*-*space diffusion equation*, I, Fractional Calculus and Applied Analysis **8** (2005). Search in Google Scholar

[36] M. Karkulik and J. M. Melenk, 𝓗-*matrix approximability of inverses of discretizations of the fractional Laplacian*, Advances in Computational Mathematics (2019). Search in Google Scholar

[37] K. Kunisch and S. Volkwein, *Galerkin proper orthogonal decomposition methods for parabolic problems*, Numerische Mathematik **90** (2001), 117–148. Search in Google Scholar

[38] M. Kwaśnicki, *Ten equivalent definitions of the fractional Laplace operator*, Fractional Calculus and Applied Analysis **20** (2015). Search in Google Scholar

[39] J.-L. Lions and E. Magenes, *Non*-*homogeneous boundary value problems and applications*, Springer-Verlag, New York, 1972. Search in Google Scholar

[40] A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. Meerschaert, M. Ainsworth, and G. Karniadakis, *What is the fractional Laplacian? A comparative review with new results*, Journal of Computational Physics **404** (2020). Search in Google Scholar

[41] Y. Maday, A. T. Patera, and G. Turinici, *Global a priori convergence theory for reduced*-*basis approximations of single*-*parameter symmetric coercive elliptic partial differential equations*, Comptes Rendus Mathematique **335** (2002), no. 3, 289 – 294. Search in Google Scholar

[42] S. Margenov, P. Marinov, R. Lazarov, S. Harizanov, and Y. Vutov, *Optimal solvers for linear systems with fractional powers of sparse SPD matrices*, Numerical Linear Algebra with Applications **25** (2018). Search in Google Scholar

[43] D. Meidner, J. Pfefferer, K. Schürholz, and B. Vexler, *hp*-*finite elements for fractional diffusion*, SIAM Journal on Numerical Analysis **56** (2017). Search in Google Scholar

[44] J. M. Melenk and A. Rieder, *hp*-*FEM for the fractional heat equation*, IMA Journal of Numerical Analysis (2020). Search in Google Scholar

[45] R. H. Nochetto, E. Otárola, and A. J. Salgado, *A PDE approach to fractional diffusion in general domains: A priori error analysis*, Found. of Comput. Math. **15** (2015), 733–791. Search in Google Scholar

[46] J. E. Pasciak, S. Margenov, P. Marinov, R. Lazarov, and S. Harizanov, *Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation*, Journal of Computational Physics **408** (2020). Search in Google Scholar

[47] J. Peetre, *On the theory of interpolation spaces*, (1963). Search in Google Scholar

[48] P. Perdikaris and G. E. Karniadakis, *Fractional*-*order viscoelasticity in onedimensional blood flow models*, Annals of Biomedical Engineering **42** (2014), 1012–1023. Search in Google Scholar

[49] A. Quarteroni, A. Manzoni, and F. Negri, *Reduced basis methods for partial differential equations*, Springer International Publishing, 2016. Search in Google Scholar

[50] G. Rozza, D. B. P. Huynh, and A. T. Patera, *Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations*, Archives of Computational Methods in Engineering **15** (2008). Search in Google Scholar

[51] J. Schöberl, *Netgen an advancing front 2D/3D-mesh generator based on abstract rules*, Computing and Visualization in Science **1** (1997), 41–52. Search in Google Scholar

[52] J. Schöberl, *C++11 implementation of Finite Elements in NGSolve*, (2014). Search in Google Scholar

[53] J. Sprekels and E. Valdinoci, *A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation*, SIAM J. Control and Optimization **55** (2016), 70–93. Search in Google Scholar

[54] P. R. Stinga and J. L. Torrea, *Extension problem and Harnack’s inequality for some fractional operators*, Communications in Partial Differential Equations **35** (2010), 2092–2122. Search in Google Scholar

[55] L. Tartar, *An introduction to sobolev spaces and interpolation spaces*, vol. 3, Springer-Verlag, Berlin, 2007. Search in Google Scholar

[56] H. Triebel, *Interpolation theory, function spaces, differential operators*, North- Holland Pub., 1978. Search in Google Scholar

[57] P. N. Vabishchevich, *Numerically solving an equation for fractional powers of elliptic operators*, Journal of Computational Physics **282** (2015), 289 – 302. Search in Google Scholar

[58] D. R. Witman, M. Gunzburger, and J. Peterson, *Reduced*-*order modeling for nonlocal diffusion problems*, International Journal for Numerical Methods in Fluids (2016). Search in Google Scholar

[59] E. I. Zolotarëv, *Collected works*, St.-Petersburg Academy of Sciences (1877). Search in Google Scholar

## 6 Appendix

## Proof

*Proof of*Theorem 2.2. It suffices to show that

There holds

Let *u _{k}* := 〈

*u*,

*φ*〉

_{k}_{0}to deduce from Lemma 2.1

which proves (24) and concludes the proof.□

## Proof

*Proof of*Theorem 2.3. One observes that for any *F* ∈ 𝓥_{0}

from which we conclude that _{−1}-orthonormal system of eigenfunctions. Since

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

## Proof

*Proof of*Theorem 2.5. Due to

and Theorem 2.3, there holds

proving the first equality in (8). The second one follows by means of (3). Furthermore, one observes

confirming the first equality in (9). The latter is a consequence of (4). The remainder follows as *F* ∈ 𝓥_{0}.□

**Published Online:**2021-03-09

© 2021 Walter de Gruyter GmbH, Berlin/Boston