Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise (Q766225)

The authors consider the abstract stochastic Cauchy problem \[ dX(t)+AX(t)dt=BdW(t),\;t>0;\;X(0)=X_{0}, \] in a Hilbert space \(H\;(X\in H)\) with a \(U\)-valued Wiener process \(W(t),\;U\) is a Hilbert space, \(-A\) is the generator of a strongly continuous semigroup. Let a finite element approximation \(X_{h}(t)\) have strong convergence of an order \(\beta:\) \[ (E\parallel X_{h}(t)-X(t)\parallel^{2})^{1/2}=O(h^{\beta}). \] The authors consider the weak error at \(T>0:\) \[ e_{h}(T):=E(G(X_{h}(T)))-E(G(X(T))), \] where \(G\) is a test function. They prove that under appropriate conditions \[ e_{h}(T)=O(h^{2\beta}\mid\log h\mid). \] The results are applied to the stochastic heat, linearized Cahn-Hilliard, and wave equations.