Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

@article{Kovcs2012WeakCO,
  title={Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes},
  author={Mih{\'a}ly Kov{\'a}cs and Stig Larsson and Fredrik Lindgren},
  journal={BIT Numerical Mathematics},
  year={2012},
  volume={53},
  pages={497-525}
}
We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It… Expand
Weak Convergence of Finite Element Approximations of Linear Stochastic Evolution Equations with Additive Lévy Noise
TLDR
An abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive L\'evy noise is presented and the weak rate of convergence is found to be twice the strong rate. Expand
Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
TLDR
Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate and this extends earlier work by one of the authors to the semilinear setting. Expand
Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise
A unified approach is given for the analysis of the weak error of spatially semidiscrete finite element methods for linear stochastic partial differential equations driven by additive noise. An errorExpand
Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
TLDR
The weak rate of convergence of approximate solutions of the nonlinear stochastic heat equation, when discretized in space by a standard finite element method, is found, essentially twice the rate of strong convergence. Expand
Weak convergence for a stochastic exponential integrator and finite element discretization of stochastic partial differential equation with multiplicative & additive noise
We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weakExpand
On weak and strong convergence of numerical approximations of stochastic partial differential equations
This thesis is concerned with numerical approximation of linear stochastic partial differential equations driven by additive noise. In the first part, we develop a framework for the analysis ofExpand
Error estimates of finite element method for semilinear stochastic strongly damped wave equation
In this paper, we consider a semi-linear stochastic strongly damped wave equation driven by additive Gaussian noise. Following a semigroup framework, we establish existence, uniqueness and space-timeExpand
An Exponential Integrator Scheme for Time Discretization of Nonlinear Stochastic Wave Equation
  • Xiaojie Wang
  • Mathematics, Computer Science
  • J. Sci. Comput.
  • 2015
TLDR
This paper provides a weak error analysis, which does not rely on the Malliavin calculus, and shows that the proposed method achieves higher convergence rates than the implicit Euler and Crank–Nicolson time discretizations. Expand
An accelerated exponential time integrator for semi-linear stochastic strongly damped wave equation with additive noise ☆
Abstract This paper is concerned with the strong approximation of a semi-linear stochastic wave equation with strong damping, driven by additive noise. Based on a spatial discretization performed byExpand
Weak convergence for a stochastic exponential integrator and finite element discretization of SPDE for multiplicative \& additive noise
We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weakExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 31 REFERENCES
Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise
A unified approach is given for the analysis of the weak error of spatially semidiscrete finite element methods for linear stochastic partial differential equations driven by additive noise. An errorExpand
Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation
A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based onExpand
Finite-element approximation of the linearized Cahn–Hilliard–Cook equation
The linearized Cahn–Hilliard–Cook equation is discretized in the spatial variables by a standard finite-element method. Strong convergence estimates are proved under suitable assumptions on theExpand
Weak approximation of stochastic partial differential equations: the nonlinear case
TLDR
It is proved that as it is often the case, the weak order of convergence is twice the strong order and Malliavin calculus is a key ingredient in this proof. Expand
Finite element approximation of the linear stochastic Cahn-Hilliard equation
The linearized Cahn-Hilliard-Cook equation is discretized in the spatial variables by a standard finite element method. Strong convergence estimates are proved under suitable assumptions on theExpand
Finite Element Approximation of the Linear Stochastic Wave Equation with Additive Noise
TLDR
Semidiscrete finite element approximation of the linear stochastic wave equation with additive noise with minimal regularity assumptions is studied in a semigroup framework to prove strong convergence estimates for the Stochastic problem. Expand
Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise
AbstractWe study the approximation of the distribution of XT, where (Xt)t ∈ [0, T] is a Hilbert space valued stochastic process that solves a linear parabolic stochastic partial differential equationExpand
Approximation for Semilinear Stochastic Evolution Equations
We investigate the approximation by space and time discretization of quasi linear evolution equations driven by nuclear or space time white noise. An error bound for the implicit Euler, the explicitExpand
Semidiscrete and single step fully discrete approximations for second order hyperbolic equations
Fimte element approximations are analysed, for initial boundary value problems far second ox&ezJiyperboUc équations For both semidiscrete andfully discrete schémas, optimal order rate o f convergenceExpand
Weak and Strong Order of Convergence of a Semidiscrete Scheme for the Stochastic Nonlinear Schrodinger Equation
In this article we analyze the error of a semidiscrete scheme for the stochastic nonlinear Schrodinger equation with power nonlinearity. We consider supercritical or subcritical nonlinearity and theExpand
...
1
2
3
4
...