Stochastic Exponential Integrators for a Finite Element Discretization of SPDEs. (arXiv:1005.5315v1 [math.NA])

Stochastic Exponential Integrators for a Finite Element Discretization of SPDEs. (arXiv:1005.5315v1 [math.NA]): "

We consider the numerical approximation of general semilinear parabolic
stochastic partial differential equations (SPDEs) driven by additive space-time
noise. In contrast to the standard time stepping methods which uses basic
increments of the noise and the approximation of the exponential function by a
rational fraction, we introduce a new scheme, designed for finite elements,
finite volumes or finite differences space discretization, similar to the
schemes in \cite{Jentzen3,Jentzen4} for spectral methods and \cite{GTambue} for
finite element methods. We use the projection operator, the smoothing effect of
the positive definite self-adjoint operator and linear functionals of the noise
in Fourier space to obtain higher order approximations. We consider noise that
is white in time and either in $H^1$ or $H^2$ in space and give convergence
proofs in the mean square $L^{2}$ norm for a diffusion reaction equation and in
mean square $ H^{1}$ norm in the presence of an advection term. For the
exponential integrator we rely on computing the exponential of a non-diagonal
matrix. In our numerical results we use two different efficient techniques: the
real fast \Leja points and Krylov subspace techniques. We present results for a
linear reaction diffusion equation in two dimensions as well as a nonlinear
example of two-dimensional stochastic advection diffusion reaction equation
motivated from realistic porous media flow.

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