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The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds. (English) Zbl 1208.65017

The authors consider the pathwise numerical approximation of the stochastic evolution equation

$\begin{array}{cc}& du\left(t\right)=\left[Au\left(t\right)+F\left(u\left(t\right)\right)\right]dt+dW\left(t\right);t\ge 0;\hfill \\ & u\left(0\right)={u}_{0};\hfill \end{array}$

on the Hilbert space $H={L}_{2}\left({\left[a;b\right]}^{d}\right)$, where $A$ is the generator of an analytic semigroup $H$, ${u}_{0}\in D\left(A\right)$, $W$ is an $H$-valued $Q$-Wiener process, the mapping $F$ is nonlinear with bounded first and second derivatives. They also assume that $A$ and $Q$ have common eigenfunctions. However, the covariance operator $Q$ needs not to be a trace-class operator.

To treat this equation numerically, the authors discretise in space by a Galerkin method and in time by a stochastic exponential integrator. They estimate the rates of convergence for both $p$th-mean and pathwise errors of approximations. It appears that for spatially regular noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces. The results are illustrated by the two-dimensional Allen–Cahn equation.

##### MSC:
 65C30 Stochastic differential and integral equations 60H15 Stochastic partial differential equations 65M12 Stability and convergence of numerical methods (IVP of PDE) 65M15 Error bounds (IVP of PDE) 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 60H35 Computational methods for stochastic equations 35R60 PDEs with randomness, stochastic PDE