
Numerical solution of timedependent problems with fractional power elliptic operator
An unsteady problem is considered for a spacefractional equation in a b...
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The Best Uniform Rational Approximation: Applications to Solving Equations Involving Fractional powers of Elliptic Operators
In this paper we consider one particular mathematical problem of this la...
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Reduced Sum Implementation of the BURA Method for Spectral Fractional Diffusion Problems
The numerical solution of spectral fractional diffusion problems in the ...
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Pseudospectral roaming contour integral methods for convectiondiffusion equations
We generalize ideas in the recent literature and develop new ones in ord...
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Outlier removal for isogeometric spectral approximation with the optimallyblended quadratures
It is wellknown that outliers appear in the highfrequency region in th...
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Some methods for solving equations with an operator function and applications for problems with a fractional power of an operator
Several applied problems are characterized by the need to numerically so...
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Elliptic polytopes and invariant norms of linear operators
We address the problem of constructing elliptic polytopes in R^d, which ...
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A Survey on Numerical Methods for Spectral SpaceFractional Diffusion Problems
The survey is devoted to numerical solution of the fractional equation A^α u=f, 0 < α <1, where A is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝ^d. The operator fractional power is a nonlocal operator and is defined through the spectrum. Due to growing interest and demand in applications of subdiffusion models to physics and engineering, in the last decade, several numerical approaches have been proposed, studied, and tested. We consider discretizations of the elliptic operator A by using an Ndimensional finite element space V_h or finite differences over a uniform mesh with N grid points. The numerical solution of this equation is based on the following three equivalent representations of the solution: (1) DunfordTaylor integral formula (or its equivalent Balakrishnan formula), (2) extension of the a second order elliptic problem in Ω× (0,∞)⊂ℝ^d+1 (with a local operator) or as a pseudoparabolic equation in the cylinder (x,t) ∈Ω× (0,1), (3) spectral representation and the best uniform rational approximation (BURA) of z^α on [0,1]. Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of A^α. In this paper we present the main ideas of these methods and the corresponding algorithms, discuss their accuracy, computational complexity and compare their efficiency and robustness.
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