Abstract
The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators. Such discrete schemes can be used in the cases of non-constant elliptic operators, non-uniform space meshes and general space domains. The stability results are proved for all algorithms and the accuracy of obtained approximations is estimated by solving well-known test problems. A modification of the second order splitting scheme is presented, it combines the splitting method to solve locally the nonlinear subproblem and the AAA algorithm to solve the nonlocal diffusion subproblem. Results of computational experiments are presented and analyzed.
Highlights
Schemes for Numerical Solution ofIn recent decades fractional differential equations proved to be important techniques for modeling diffusive type processes when an anomalous diffusion is important
In general case the implementation of the constructed discrete schemes requires to use some approximations of the operators with fractional powers of discrete elliptic operators
The iterations are terminated when the nonlinear residual is sufficiently small. This method proved to be very efficient in solving stationary equations for fractional power elliptic operators
Summary
In recent decades fractional differential equations proved to be important techniques for modeling diffusive type processes when an anomalous diffusion is important. It is important to note, that the given spectral algorithm can be used for practical computations to solve parabolic type problems with nonlocal operator Aαh. In this paper we will use this method widely as an essential part of three algorithms proposed to solve parabolic problems with fractional power elliptic operators. We mention papers [40,41], where efficient discrete schemes are proposed to solve nonstationary applied problems with fractional derivatives or different definitions of fractional powers of elliptic operators. Such techniques are important in the case of fractional powers of elliptic operators, when the spectral definition is used.
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