11,723 research outputs found
Finite element approximation of the linear stochastic wave equation with additive noise
Semidiscrete finite element approximation of the linear stochastic wave equation (LSWE) with additive noise is studied in a semigroup framework. Optimal error estimates for the deterministic problem are obtained under minimal regularity assumptions. These are used to prove strong convergence estimates for the stochastic problem. The theory presented here applies to multidimensional domains and spatially correlated noise. Numerical examples illustrate the theory
Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise
A fully discrete approximation of the semi-linear stochastic wave equation
driven by multiplicative noise is presented. A standard linear finite element
approximation is used in space and a stochastic trigonometric method for the
temporal approximation. This explicit time integrator allows for mean-square
error bounds independent of the space discretisation and thus do not suffer
from a step size restriction as in the often used St\"ormer-Verlet-leap-frog
scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift
of the expected value of the energy of the problem). Numerical experiments are
presented and confirm the theoretical results
Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes
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 is found that the rate of weak
convergence is twice that of strong convergence. Furthermore, in contrast to
the parabolic case, higher order schemes in time, such as the Crank-Nicolson
scheme, are worthwhile to use if the solution is not very regular. Finally we
apply the theory to parabolic equations and detail a weak error estimate for
the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic
heat equation
Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
We find the weak rate of convergence of the spatially semidiscrete finite
element approximation of the nonlinear stochastic heat equation. Both
multiplicative and additive noise is considered under different assumptions.
This extends an earlier result of Debussche in which time discretization is
considered for the stochastic heat equation perturbed by white noise. It is
known that this equation has a solution only in one space dimension. In order
to obtain results for higher dimensions, colored noise is considered here,
besides white noise in one dimension. Integration by parts in the Malliavin
sense is used in the proof. The rate of weak convergence is, as expected,
essentially twice the rate of strong convergence.Comment: 19 page
- …