Semidiscrete finite element approximation of the linear stochastic wave equation (LSWE) with additive noise is studied in a semigroup framework. Optimal error estimates for the deterministic problem are obtained under minimal regularity assumptions. These are used to prove strong convergence estimates for the stochastic problem. The theory presented here applies to multidimensional domains and spatially correlated noise. Numerical examples illustrate the theory. 

Motivated by fractional derivative models in viscoelasticity, a class of semilinear stochastic Volterra integro-differential equations, and their deterministic counterparts, are considered. A generalized exponential Euler method, named here the Mittag--Leffler Euler integrator, is used for the temporal discretization, while the spatial discretization is performed by the spectral Galerkin method. The temporal rate of strong convergence is found to be (almost) twice compared to when the backward Euler method is used together with a convolution quadrature for time discretization. Numerical experiments that validate the theory are presented.  

The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our error bounds for integro-differential equations. The order conditions are further used for constructing numerical methods. The convergence analysis is performed in a Hilbert space setting, where the smoothing effect of the resolvent family is heavily used. For the linear case, we derive the order conditions for general order p and prove convergence of order p, whenever these conditions are satisfied. In the semilinear case, we consider in addition spatial discretization by a spectral Galerkin method, and we require locally Lipschitz continuous nonlinearities. We derive the order conditions for orders one and two, construct methods satisfying these conditions and prove their convergence. Finally, some numerical experiments illustrating our theoretical results are given.