Numerical analysis of semilinear stochastic evolution equations in Banach spaces

Hausenblas, Erika

doi:10.1016/s0377-0427(02)00483-1

Cited by 70 publications

(58 citation statements)

References 25 publications

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“…Hausenblas investigated the discretization error of semilinear stochastic evolution equations in L p -spaces, Banach spaces and quasi linear evolution equations driven by nuclear or space time white noise in [8,9]. Gyöngy and Shardlow in [18,7] apply finite differences in order to approximate the mild solution of parabolic SPDEs driven by space-time white noise.…”

Section: Introductionmentioning

confidence: 99%

Full discretization of the stochastic Burgers equation with correlated noise

Blömker

Kamrani

Hosseini

2013

IMA Journal of Numerical Analysis

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The main purpose of this paper is to investigate the spectral Galerkin method for spatial discretization. We combine it with the method introduced by Kloeden, Jentzen & Winkel in [12] for temporal discretization of stochastic partial differential equations and study pathwise convergence. We consider the case of colored noise, instead of the usual space-time white noise that was used before for the spatial discretization. The rate of convergence in uniform topology is estimated for the stochastic Burgers equation. Numerical examples illustrate the estimated convergence rate.

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Section: Introductionmentioning

confidence: 99%

Full discretization of the stochastic Burgers equation with correlated noise

Blömker

Kamrani

Hosseini

2013

IMA Journal of Numerical Analysis

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“…A partial list of further contributions includes Allen, Novosel, Zhang (1998), Davie, Gaines (2001), Du, Zhang (2002), Gyöngy (1999), Hausenblas (2002Hausenblas ( , 2003, Kloeden, Shott (2001), Lord, Rougemont (2003), Shardlow (1999), and Yan (2003aYan ( , 2003b.…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

Müller-Gronbach¹,

Ritter

2006

Found Comput Math

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Abstract. We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0, 1[ d . The error of an algorithm is defined in L 2 -sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W . For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of non-uniform time discretizations is crucial.

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“…In spite of the recent contributions in the convergence analysis of the finite element method for second order stochastic parabolic problems with additive space-time white noise (see, e.g., [1,3,14,19,25,27,29]), the order of convergence of the finite element method for the fourth order problem (1.1) is not conclusive. Also, we refer to [6] for the convergence analysis of a finite difference method, to [15,16,23] for the convergence analysis of time-discretization methods for a wide family of evolution problems that includes (1.1), while the finite element method is not among the space-discretization techniques considered in [15,16].…”

Section: Main Results Of the Papermentioning

confidence: 99%

“…Following the adopted techniques, such estimates can be also derived for the second order equations, for which, to our knowledge, there are no analogue results in the existing literature (cf. [3,15,16,25,27,29]). -A first impression is that, estimates (1.14) and (1.16), are qualitatively better than the estimates (1.15) and (1.17), respectively, because no negative powers of Δt or Δτ appear.…”

Section: Main Results Of the Papermentioning

confidence: 99%

Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise

Kossioris¹,

Zouraris²

2010

ESAIM: M2AN

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Abstract.We consider an initial and Dirichlet boundary value problem for a fourth-order linear stochastic parabolic equation, in one space dimension, forced by an additive space-time white noise. Discretizing the space-time white noise a modelling error is introduced and a regularized fourth-order linear stochastic parabolic problem is obtained. Fully-discrete approximations to the solution of the regularized problem are constructed by using, for discretization in space, a Galerkin finite element method based on C 0 or C 1 piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates for the modelling error and for the approximation error to the solution of the regularized problem.Mathematics Subject Classification. 65M60, 65M15, 65C20.

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Numerical analysis of semilinear stochastic evolution equations in Banach spaces

Cited by 70 publications

References 25 publications

Full discretization of the stochastic Burgers equation with correlated noise

Full discretization of the stochastic Burgers equation with correlated noise

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise

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