Numerical Approximation of Parabolic SPDE's

Convergence theory for numerical schemes to approximate solutions of stochastic parabolic equations of the form

$$ du + A(u) \, dt = f \, dt + g \, dW, \qquad u(0)=u^0,$$

will be reviewed. Here $u$ is a random variable taking values in a function space $U$, $A:U \rightarrow U'$ is partial differential operator, $W = \{W_t\}_{t \geq 0}$ a Wiener process, and $f$, $g$, and $u^0$ are data. This talk will illustrate how techniques from stochastic analysis and numerical partial differential equations can be combined to obtain a realization of the Lax--Richtmeyer meta--theorem 

A numerical scheme converges if (and only if) it is stable and consistent.

Structural properties of the partial differential operator(s) and probabilistic methods will be developed to establish stability and a version of Donsker's theorem for discrete processes in the dual space $U'$.

This is joint work with M. Ondrejat (Prague, CZ) and A. Prohl (Tuebingen, DE).