Domain decomposition strategies for the stochastic heat equation

Carelli, Erich; Müller, Alexander; Prohl, Andreas

doi:10.1080/00207160.2012.720977

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…Recently stochastic convection-diffusion problems have attracted considerable interest [2,6,7,8,17,19,26]. To obtain efficient numerical discretizations adaptivity is mandatory.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for non-stationary non-linear convection–diffusion equations

Verfürth

2018

Calcolo

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Motivated by stochastic convection-diffusion problems we derive a posteriori error estimates for non-stationary non-linear convection-diffusion equations acting as a deterministic paradigm. The problem considered here neither fits into the standard linear framework due to its non-linearity nor into the standard non-linear framework due to the lacking differentiability of the non-linearity. Particular attention is paid to the interplay of the various parameters controlling the relative sizes of diffusion, convection, reaction, and non-linearity (noise).(2.1) Here, Ω ⊂ R d , d ∈ {2, 3}, is a bounded polyhedral domain with Lipschitz boundary Γ. The final time T is arbitrary, but kept fixed in what follows. We assume that the data satisfy the following conditions (compare [21, §3] and [25, §6.2]):(A1) ε > 0, ν ≥ 0,

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“…Recently stochastic convection-diffusion problems have attracted considerable interest [2,6,7,8,17,19,26]. To obtain efficient numerical discretizations adaptivity is mandatory.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for non-stationary non-linear convection–diffusion equations

Verfürth

2018

Calcolo

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“…Despite this fact we believe that our paper is still interesting as we are able to treat a class of 2D stochastic nonlinear heat equations with locally Lipschitz coefficients and we are not aware of results similar to ours. In fact, most of results related to stochastic heat equations are either about 1D model, or d-dimensional, d ∈ {1, 2, 3}, models with globally Lipschitz coefficients and deal with weak convergence or convergence in weaker norm, see for instance [38,16,24,1]. This paper is organized as follows: in Section 2, we introduce the necessary notations and the standing assumptions that will be used in the present work.…”

Section: Introductionmentioning

confidence: 99%

“…The literature on numerical analysis for SPDEs is now very extensive. Without being exhaustive, we only cite amongst other the recent papers [38,16,24,1,15], the excellent review paper [33] and references therein. Most of the literature deals with the stochastic heat equations with globally Lipschitz nonlinearities, but there are also several papers that treat abstract stochastic evolution equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

Bessaih

Hausenblas

Randrianasolo

et al. 2018

Journal of Computational and Applied Mathematics

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The present paper is devoted to the numerical approximation of an abstract stochastic nonlinear evolution equation in a separable Hilbert space H. Examples of equations which fall into our framework include the GOY and Sabra shell models and a class of nonlinear heat equations. The spacetime numerical scheme is defined in terms of a Galerkin approximation in space and a semi-implicit Euler-Maruyama scheme in time. We prove the convergence in probability of our scheme by means of an estimate of the error on a localized set of arbitrary large probability. Our error estimate is shown to hold in a more regular space V β ⊂ H with β ∈ [0, 1 4 ) and that the explicit rate of convergence of our scheme depends on this parameter β.

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Preface: ‘Recent advances in the numerical approximation of stochastic partial differential equations’

Kloeden¹,

Ritter

2012

International Journal of Computer Mathematics

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Preface: 'Recent advances in the numerical approximation of stochastic partial differential equations' Stochastic partial differential equations (SPDEs) of evolutionary type are used to model continuous-time random dynamics in infinite-dimensional state spaces. They have been an active area of research for a number of decades and a deep well-developed theory is now available. A comparable numerical analysis of such equations was delayed by various technical difficulties, but there has been considerable progress in recent years. The NSF/CBMS Regional Conference in the Mathematical Sciences on Recent Advances in the Numerical Approximation of Stochastic Partial Differential Equations, which was held at the

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Domain decomposition strategies for the stochastic heat equation

Cited by 3 publications

References 9 publications

A posteriori error estimates for non-stationary non-linear convection–diffusion equations

A posteriori error estimates for non-stationary non-linear convection–diffusion equations

Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

Preface: ‘Recent advances in the numerical approximation of stochastic partial differential equations’

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