Discrete maximal regularity for abstract Cauchy problems

Kemmochi, Tomoya

doi:10.4064/sm8495-7-2016

Cited by 19 publications

(34 citation statements)

References 25 publications

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“…These estimates are sharp with respect to the regularity of the solution in Theorem 3.1 (up to a logarithmic factor ℓ h ), and are confirmed by the numerical experiments in Section 6. Besides, we show how to simplify the analysis of nonlinear problems by applying the fractional-type discrete maximal ℓ p -regularity established in [17], an extension of the discrete maximal ℓ p -regularity of standard parabolic equations [18,21,25], which has been applied to numerical analysis of nonlinear parabolic equations in the literature [1,2,22].…”

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confidence: 99%

Numerical Analysis of Nonlinear Subdiffusion Equations

Jin¹,

Li²,

Zhou³

2018

SIAM J. Numer. Anal.

210

133

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Abstract. We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order α ∈ (0, 1) in time. It relies on three technical tools: a fractional version of the discrete Grönwall-type inequality, discrete maximal regularity, and regularity theory of nonlinear equations. We establish a general criterion for showing the fractional discrete Grönwall inequality, and verify it for the L1 scheme and convolution quadrature generated by BDFs. Further, we provide a complete solution theory, e.g., existence, uniqueness and regularity, for a time-fractional diffusion equation with a Lipschitz nonlinear source term. Together with the known results of discrete maximal regularity, we derive pointwise L 2 (Ω) norm error estimates for semidiscrete Galerkin finite element solutions and fully discrete solutions, which are of order O(h 2 ) (up to a logarithmic factor) and O(τ α ), respectively, without any extra regularity assumption on the solution or compatibility condition on the problem data. The sharpness of the convergence rates is supported by the numerical experiments.Keywords: nonlinear fractional diffusion equation, discrete fractional Grönwall inequality, L1 scheme, convolution quadrature, error estimate 1. Introduction. Time-fractional parabolic partial differential equations (PDEs) have been very popular for modeling anomalously slow transport processes in the past two decades. These models are commonly referred to as fractional diffusion or subdiffusion. At a microscopic level, the underlying stochastic process is continuous time random walk [32]. So far they have been successfully applied in a broad range of diversified research areas, e.g., thermal diffusion in fractal domains [35], flow in highly heterogeneous aquifer [6] and single-molecular protein dynamics [20], just to name a few. Hence, the rigorous numerical analysis of such problems is of great practical importance. For the linear problem, various efficient time stepping schemes have been proposed, which include mainly two classes: L1 type schemes and convolution quadrature (CQ).L1 type schemes approximate the fractional derivative by replacing the integrand with its piecewise polynomial interpolation [24,26,37,3] and thus generalize the classical finite difference method. The piecewise linear case has a local truncation error O(τ 2−α ) for sufficiently smooth solution, where τ denotes the time step size. See also [31,33] for the discontinuous Galerkin method. CQ is a flexible framework introduced by Lubich [27,28] for constructing high-order time discretization methods for approximating fractional derivatives. It approximates the fractional derivative in the Laplace domain and automatically inherits the stability property of general linear multistep methods. See [10,39,40,16] for CQ type schemes. Optimal error estimates have been derived for both spatially semidiscrete and fully discrete schemes, including problems with nonsmooth data [10,14,31,16]....

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confidence: 99%

Numerical Analysis of Nonlinear Subdiffusion Equations

Jin¹,

Li²,

Zhou³

2018

SIAM J. Numer. Anal.

210

133

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show abstract

“…Starting with the work of Blunck, the existence and uniqueness of solutions for discrete systems that belong to the Lebesgue space of vector‐valued sequences began to be considered by many authors. () Some of the studies correspond to a numerical point of view . However, none of them have considered causal solutions, ie, solutions with domain on

double-struckZ .

On the other hand, some abstract models that are less general than ours have been recently reported in the literature and analyzed from diverse perspectives.…”

Section: Introductionmentioning

confidence: 97%

“…[14][15][16] Other interesting contributions are due to Ferreira, 17 Holm, 18 Kovács, Li, and Lubich, 19 Dassios,20,21 Wu, Baleanu et al, [22][23][24][25] and Tarasov et al [26][27][28] Starting with the work of Blunck, 29 the existence and uniqueness of solutions for discrete systems that belong to the Lebesgue space of vector-valued sequences began to be considered by many authors. [30][31][32][33] Some of the studies correspond to a numerical point of view. 19 However, none of them have considered causal solutions, ie, solutions with domain on Z.…”

Section: Introductionmentioning

confidence: 99%

Lebesgue regularity for differential difference equations with fractional damping

Lizama

Murillo‐Arcila

Leal

2018

Math Methods in App Sciences

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We provide necessary and sufficient conditions for the existence and uniqueness of solutions belonging to the vector‐valued space of sequences ℓpfalse(double-struckZ,Xfalse) for equations that can be modeled in the form normalΔαu(n)+λnormalΔβu(n)=Au(n)+G(u)(n)+f(n),n∈Z,α,β>0,λ≥0, where X is a Banach space, f∈ℓpfalse(double-struckZ,Xfalse), A is a closed linear operator with domain D(A) defined on X, and G is a nonlinear function. The operator Δγ denotes the fractional difference operator of order γ>0 in the sense of Grünwald‐Letnikov. Our class of models includes the discrete time Klein‐Gordon, telegraph, and Basset equations, among other differential difference equations of interest. We prove a simple criterion that shows the existence of solutions assuming that f is small and that G is a nonlinear term.

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“…In recent years, the study of discrete analogues of the maximal regularity has been attracting attention for deterministic partial differential equations [1,4,7,13,14,15,20]; to the best of the author's knowledge, corresponding properties of numerical methods for stochastic PDEs have not been addressed in the literature.…”

Section: Introductionmentioning

confidence: 99%

Discrete maximal regularity of an implicit Euler–Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations

Kazashi¹

2018

Electron. Commun. Probab.

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An implicit Euler-Maruyama method with non-uniform step-size applied to a class of stochastic partial differential equations is studied. A spectral method is used for the spatial discretization and the truncation of the Wiener process. A discrete analogue of maximal L 2 -regularity of the scheme and the discretised stochastic convolution is established, which has the same form as their continuous counterpart.

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Discrete maximal regularity for abstract Cauchy problems

Cited by 19 publications

References 25 publications

Numerical Analysis of Nonlinear Subdiffusion Equations

Numerical Analysis of Nonlinear Subdiffusion Equations

Lebesgue regularity for differential difference equations with fractional damping

Discrete maximal regularity of an implicit Euler–Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations

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