# 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

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#### References

SHOWING 1-10 OF 31 REFERENCES

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise

- Mathematics
- 2012

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 error… Expand

Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation

- Mathematics
- 1992

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 on… Expand

Finite-element approximation of the linearized Cahn–Hilliard–Cook equation

- Mathematics
- 2011

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 the… Expand

Weak approximation of stochastic partial differential equations: the nonlinear case

- Mathematics, Computer Science
- Math. Comput.
- 2011

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

- Mathematics
- 2009

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 the… Expand

Finite Element Approximation of the Linear Stochastic Wave Equation with Additive Noise

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2010

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

- Mathematics
- 2009

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 equation… Expand

Approximation for Semilinear Stochastic Evolution Equations

- Mathematics
- 2003

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 explicit… Expand

Semidiscrete and single step fully discrete approximations for second order hyperbolic equations

- Mathematics
- 1979

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 convergence… Expand

Weak and Strong Order of Convergence of a Semidiscrete Scheme for the Stochastic Nonlinear Schrodinger Equation

- Mathematics
- 2006

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 the… Expand