Numerical Solution of Time-Dependent Problems with Fractional Power Elliptic Operator-Reference-Cited by-同舟云学术

Numerical Solution of Time-Dependent Problems with Fractional Power Elliptic Operator

Published:2017-08-17 Issue:1 Volume:18 Page:111-128
ISSN:1609-9389
Container-title:Computational Methods in Applied Mathematics
language:en
Short-container-title:

Author:

Vabishchevich Petr N.¹

Affiliation:

1. Nuclear Safety Institute , Russian Academy of Sciences , 52, B. Tulskaya,115191 Moscow ; and Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow , Russia

Abstract

Abstract 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.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Computational Mathematics,Numerical Analysis

Reference45 articles.

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.

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.

5. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 2008.

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