Nonsmooth data error estimates for damped single step methods for parabolic equations in Banach space

Hansbo, Anita

doi:10.1007/s100920050024

Cited by 30 publications

(20 citation statements)

References 13 publications

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“…Rannacher [31], and in the final form Hansbo [15] proposed a stabilization technique for unstable rational approximations by first applying a stable lower order approximation with r(∞) = 0 and then combining the smoothing property of the lower order scheme with the improved accuracy of the higher order approximation scheme. The stabilization of rational approximation schemes for non-analytic strongly continuous semigroups was investigated by McAllister and Neubrander in [28].…”

Section: Lemmamentioning

confidence: 99%

“…For u ∈ C ub (Σ θ , X) ∩ H(Σ θ , X) the results can be improved by using Hansbo's stabilization methods [15]; for u, u (1) ∈ C b (R + , X) the results can be improved by stabilizing the Crank-Nicolson scheme using the methods in [28] (see [29]). …”

Section: Restricted Padé Inversion Of the Laplace Transformmentioning

confidence: 99%

“…They are, therefore, applicable to parabolic and hyperbolic problems alike. The inversion methods are derived from the theory of rational approximation methods for strongly continuous operator semigroups using results due to Hersh and Kato [16], Brenner and Thomée [4], Larsson, Thomée and Wahlbin [27], Hansbo [15], and Crouzeix, Larsson, Piskarev and Thomée [8] (see also [19], [22], and [34]). …”

Section: Introductionmentioning

confidence: 99%

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Rational inversion of the Laplace transform

2012

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Abstract. We introduce Post-Widder type inversion methods for the Laplace transform based on A-stable rational approximations of the exponential function of order m ≥ 1. It is shown that bounded, continuous functions u can be approximated in terms of their Laplace transformsû by expressionswhere the coefficients a j,k , b k are independent of u and Re(b k ) > 0. If u is analytic, uniformly continuous, and bounded in a sectorial region containing the half-line (0, ∞), then the mathematical approximation error is bounded by C 1 n m u ∞ (with the constant C being independent of t and u); if u is (m+1)-times continuously differentiable and bounded, then the error is bounded by Ct 1 n m u (m+1) ∞; if u is continuously differentiable and bounded, then the error is bounded by Ct 1 n β u ∞ for some 1 2 ≤ β < 1. In particular, if u is sufficiently smooth, then n, the order of the derivatives ofû, can be kept low by taking rational approximations of the exponential function of high approximation order m. Since the results hold for Banach space valued functions, they yield efficient time-discretization methods for evolution equations of convolution type (e.g. linear first and higher order abstract Cauchy problems, inhomogeneous Cauchy problems, delay equations, Volterra and integro-differential equations, and problems that can be re-written as an abstract Cauchy problem u (t) = Au(t), u(0) = x, t ∈ [0, T ] on an appropriate state space X).

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Section: Lemmamentioning

confidence: 99%

Section: Restricted Padé Inversion Of the Laplace Transformmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rational inversion of the Laplace transform

2012

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“…The similar results in single-step methods for homogeneous parabolic problems in general Banach space have been studied, for example, by Hansbo [5,6] and Yan [12,13].…”

Section: Introductionmentioning

confidence: 63%

Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations

Yan

2005

International Journal of Mathematics and Mathematical Sciences

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A smoothing property in multistep backward difference method for a linear parabolic problem in Hilbert space has been proved, where the operator is selfadjoint, positive definite with compact inverse. By using the solutions computed by a multistep backward difference method for the parabolic problem, we introduce an approximation scheme for time derivative. The nonsmooth data error estimate for the approximation of time derivative has been obtained.

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“…However, if r(∞) = 1 (e.g., Crank-Nicolson), then the convergence can be arbitrarily slow for non-smooth x ∈ X. R. Rannacher [23], and in final form A. Hansbo [13], combined the high accuracy of the Crank-Nicolson scheme with the smoothing properties of the Backward Euler scheme to provide a remedy (for extensions, see [11]). More precisely, let r, r s be A-stable rational approximation schemes such that the high accuracy approximation r has order m, the stabilizing approximation r s order m − 1, and r s (∞) = 0.…”

Section: Stabilizationmentioning

confidence: 98%

Stabilized approximations of strongly continuous semigroups

McAllister¹,

Neubrander

2008

Journal of Mathematical Analysis and Applications

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This paper introduces stabilization techniques for intrinsically unstable, high accuracy rational approximation methods for strongly continuous semigroup. The methods not only stabilize the approximations, but improve their speed of convergence by a magnitude of up to 1/2. PreliminariesThe Lax-Richtmyer Equivalence Theorem and the Chernoff Product Formula play an important role in the analysis of approximation methods for semigroups. One of the main ingredients in both results is the stability of the approximation method under consideration. This paper deals with variants of these theorems that cover stabilization techniques for intrinsically unstable approximation methods. To put the results into perspective, recall the statement of the LaxRichtmyer Equivalence Theorem. Suppose that A generates a strongly continuous semigroup T (·) on a Banach space X and let {V (t); t ∈ [0, τ ]} be an approximation scheme of bounded linear operators with V (0) = I that satisfies the consistency conditionfor all x in a set D ⊂ D(A) (the domain of A) that is dense in X. Then it was shown by Lax and Richtmyer in 1956 [19] (with a stronger consistency condition) and in final form by Chernoff in 1974 [6] that the following statements are equivalent (see also [9]):

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Nonsmooth data error estimates for damped single step methods for parabolic equations in Banach space

Cited by 30 publications

References 13 publications

Rational inversion of the Laplace transform

Rational inversion of the Laplace transform

Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations

Stabilized approximations of strongly continuous semigroups

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