In this book we analyze the error caused by numerical schemes for the approximation of
semilinear stochastic evolution equations (SEEq) in a Hilbert space-valued setting. The
numerical schemes considered combine Galerkin finite element methods with Euler-type temporal
approximations. Starting from a precise analysis of the spatio-temporal regularity of the mild
solution to the SEEq we derive and prove optimal error estimates of the strong error of
convergence in the first part of the book. The second part deals with a new approach to the
so-called weak error of convergence which measures the distance between the law of the
numerical solution and the law of the exact solution. This approach is based on Bismut's
integration by parts formula and the Malliavin calculus for infinite dimensional stochastic
processes. These techniques are developed and explained in a separate chapter before the weak
convergence is proven for linear SEEq.
