×

Numerical solution of time-dependent problems with fractional power elliptic operator. (English) Zbl 1383.65128

Summary: An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the elliptic operator. A Padé-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power elliptic operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R11 Fractional partial differential equations
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] L. Aceto and P. Novati, Rational approximation to the fractional Laplacian operator in reaction-diffusion problems, SIAM J. Sci. Comput. 39 (2017), no. 1, 214-228.
[2] G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Numer. Anal. 55 (2017), no. 2, 472-495. · Zbl 1359.65246
[3] M. S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes and G. N. Wells, The FEniCS project version 1.5, Arch. Numer. Softw. 3 (2015), no. 100.
[4] A. Bonito and J. Pasciak, Numerical approximation of fractional powers of elliptic operators, Math. Comp. 84 (2015), no. 295, 2083-2110. · Zbl 1331.65159
[5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 2008. · Zbl 1135.65042
[6] A. Bueno-Orovio, D. Kay and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT 54 (2014), no. 4, 937-954. · Zbl 1306.65265
[7] K. Burrage, N. Hale and D. Kay, An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput. 34 (2012), no. 4, A2145-A2172. · Zbl 1253.65146
[8] C. M. Carracedo, M. S. Alix and M. Sanz, The Theory of Fractional Powers of Operators, Elsevier, Amsterdam, 2001.
[9] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Clarendon Press, New York, 1986. · Zbl 0584.73001
[10] A. Frommer, S. Güttel and M. Schweitzer, Efficient and stable Arnoldi restarts for matrix functions based on quadrature, SIAM J. Matrix Anal. Appl. 35 (2014), no. 2, 661-683. · Zbl 1309.65050
[11] W. Gautschi, Quadrature formulae on half-infinite intervals, BIT 31 (1991), no. 3, 437-446. · Zbl 0734.65007
[12] W. Gautschi, Algorithm 726: ORTHPOL - A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules, ACM Trans. Math. Software 20 (1994), no. 1, 21-62. · Zbl 0888.65013
[13] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, Oxford, 2004. · Zbl 1130.42300
[14] I. Gavrilyuk, W. Hackbusch and B. Khoromskij, Data-sparse approximation to the operator-valued functions of elliptic operator, Math. Comp. 73 (2004), no. 247, 1297-1324. · Zbl 1065.47009
[15] I. Gavrilyuk, W. Hackbusch and B. Khoromskij, Data-sparse approximation to a class of operator-valued functions, Math. Comp. 74 (2005), no. 250, 681-708. · Zbl 1066.65060
[16] V. Hernandez, J. E. Roman and V. Vidal, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Software (TOMS) 31 (2005), no. 3, 351-362. · Zbl 1136.65315
[17] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008. · Zbl 1167.15001
[18] Q. Huang, G. Huang and H. Zhan, A finite element solution for the fractional advection-dispersion equation, Adv. Water Res. 31 (2008), no. 12, 1578-1589.
[19] M. Ilic, F. Liu, I. Turner and V. Anh, Numerical approximation of a fractional-in-space diffusion equation. II: With nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal. 9 (2006), no. 4, 333-349. · Zbl 1132.35507
[20] M. Ilić, I. W. Turner and V. Anh, A numerical solution using an adaptively preconditioned Lanczos method for a class of linear systems related with the fractional Poisson equation, Int. J. Stoch. Anal. 2008 (2008), Article ID 104525. · Zbl 1162.65015
[21] B. Jin, R. Lazarov, J. Pasciak and Z. Zhou, Error analysis of finite element methods for space-fractional parabolic equations, SIAM J. Numer. Anal. 52 (2014), no. 5, 2272-2294. · Zbl 1310.65126
[22] P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer, New York, 2003. · Zbl 0953.65075
[23] M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik and P. E. Sobolevskij, Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden, 1976.
[24] A. Logg, K.-A. Mardal, G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method, Springer, Berlin, 2012.
[25] M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004), no. 1, 65-77. · Zbl 1126.76346
[26] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, 1998. · Zbl 0922.45001
[27] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, 2002. · Zbl 1027.35001
[28] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, Berlin, 1994. · Zbl 0852.76051
[29] A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, Dover Publications, Mineola, 2001. · Zbl 0976.65001
[30] Y. Saad, Numerical Methods for Large Eigenvalue Problems, SIAM, Philadelphia, 2011. · Zbl 1242.65068
[31] A. A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York, 2001. · Zbl 0971.65076
[32] A. A. Samarskii, P. P. Matus and P. N. Vabishchevich, Difference Schemes with Operator Factors, Kluwer Academic, Dordrecht, 2002. · Zbl 1018.65103
[33] E. Sousa, A second order explicit finite difference method for the fractional advection diffusion equation, Comput. Math. Appl. 64 (2012), no. 10, 3141-3152. · Zbl 1268.65118
[34] G. W. Stewart, A Krylov-Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl. 23 (2001), no. 3, 601-614. · Zbl 1003.65045
[35] B. J. Szekeres and F. Izsák, Finite element approximation of fractional order elliptic boundary value problems, J. Comput. Appl. Math. 292 (2016), 553-561. · Zbl 1327.65215
[36] C. Tadjeran, M. M. Meerschaert and H.-P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006), no. 1, 205-213. · Zbl 1089.65089
[37] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 2006. · Zbl 1105.65102
[38] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers: Applications, Higher Education Press, Beijing, 2013. · Zbl 1312.26002
[39] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers: Background and Theory, Higher Education Press, Beijing, 2013. · Zbl 1312.26002
[40] P. N. Vabishchevich, Additive Operator-Difference Schemes: Splitting Schemes, De Gruyter, Berlin, 2014.
[41] P. N. Vabishchevich, Numerically solving an equation for fractional powers of elliptic operators, J. Comput. Phys. 282 (2015), no. 1, 289-302. · Zbl 1352.65557
[42] P. N. Vabishchevich, Numerical solution of nonstationary problems for a convection and a space-fractional diffusion equation, Int. J. Numer. Anal. Model. 13 (2016), no. 2, 296-309. · Zbl 1348.65119
[43] P. N. Vabishchevich, Numerical solution of nonstationary problems for a space-fractional diffusion equation, Fract. Calc. Appl. Anal. 19 (2016), no. 1, 116-139. · Zbl 1382.65330
[44] P. N. Vabishchevich, Numerical solving unsteady space-fractional problems with the square root of an elliptic operator, Math. Model. Anal. 21 (2016), no. 2, 220-238.
[45] A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer, Berlin, 2009.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.