A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

Lin, Zhiping; Wang, Dongdong

doi:10.1007/s00466-017-1492-2

Cited by 16 publications

(7 citation statements)

References 34 publications

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“…Then we rewrite (9) in the same expression as for (1) by applying the change of temporal variables t = T − t, and solve this problem numerically by using the method (2). Hence, using change of temporal variables t = T − t second time, we establish the approximate method for the problem (9): find Θ ∈ S hk such that…”

Section: Stability For the Numerical Methods And Some Theoretical Res...mentioning

confidence: 99%

“…There are several numerical methods developed for solving RPDEs. For the space RPDE problems of first order in time, we would like to mention the works [3,5,[7][8][9][10], where the authors developed various numerical schemes that combine finite-difference and finite-element methods. For the linear space RPDE problems of second order in time, we propose a space-time bilinear finite element scheme for the numerical approximation.…”

Section: Introductionmentioning

confidence: 99%

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On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

Lai

Liu

2021

Mathematics

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In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method.

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Section: Stability For the Numerical Methods And Some Theoretical Res...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

Lai

Liu

2021

Mathematics

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“…The exponents are set to α 1 = 1.7 and α 2 = 1.9. discretization. For simplicity, we consider a uniform grid for the nodes defining the hat functions, which yields a Toeplitz matrix whose symbol is explicitly 1 given in [30]. Therefore, the same strategy used in the previous section can be used to recover a rank structured representation of the matrix, which is guaranteed to exist thanks to Theorem 3.21.…”

Section: A 2d Finite Element Discretizationmentioning

confidence: 99%

Fast Solvers for Two-Dimensional Fractional Diffusion Equations Using Rank Structured Matrices

Массеи¹,

Mazza²,

Robol³

2019

SIAM J. Sci. Comput.

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We consider the discretization of time-space diffusion equations with fractional derivatives in space and either one-dimensional (1D) or 2D spatial domains. The use of an implicit Euler scheme in time and finite differences or finite elements in space leads to a sequence of dense large scale linear systems describing the behavior of the solution over a time interval. We prove that the coefficient matrices arising in the 1D context are rank structured and can be efficiently represented using hierarchical formats (H-matrices, HODLR). Quantitative estimates for the rank of the off-diagonal blocks of these matrices are presented. We analyze the use of HODLR arithmetic for solving the 1D case and we compare this strategy with existing methods that exploit the Toeplitzlike structure to precondition the GMRES iteration. The numerical tests demonstrate the convenience of the HODLR format when at least a reasonably low number of time steps is needed. Finally, we explain how these properties can be leveraged to design fast solvers for problems with 2D spatial domains that can be reformulated as matrix equations. The experiments show that the approach based on the use of rank-structured arithmetic is particularly effective and outperforms current state of the art techniques.

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“…Various numerical discretization methods for FDE problems (e.g., finite differences, finite volumes, finite elements, spectral methods) can be found in the literature. We refer the reader to [14,19,21,24,26,37,39,40] and references therein. In the case of regular spatial domain subdivisions, the discretization matrices inherit a Toeplitz-like structure from the space-invariant property of the underlying operators that can be exploited for the design of ad hoc iterative schemes of multigrid and preconditioned Krylov type (see, e.g., [12,13,20,27,29,30]).…”

Section: Introductionmentioning

confidence: 99%

On the matrices in B-spline collocation methods for Riesz fractional equations and their spectral properties

Mazza¹,

Donatelli²,

Manni³

et al. 2021

Preprint

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In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree p, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, also in this case the given matrices are ill-conditioned both in the low and high frequencies for large p. More precisely, in the fractional scenario the symbol has a single zero at 0 of order α, with α the fractional derivative order that ranges from 1 to 2, and it presents an exponential decay to zero at π for increasing p that becomes faster as α approaches 1. This translates in a mitigated conditioning in the low frequencies and in a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B-spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B-splines. Finally, we perform a numerical study of the approximation behavior of polynomial B-spline collocation. This study suggests that, in line with non-fractional diffusion problems, the approximation order for smooth solutions in the fractional case is p + 2 − α for even p, and p + 1 − α for odd p.

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A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

Cited by 16 publications

References 34 publications

On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

Fast Solvers for Two-Dimensional Fractional Diffusion Equations Using Rank Structured Matrices

On the matrices in B-spline collocation methods for Riesz fractional equations and their spectral properties

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