Weak order for the discretization of the stochastic heat equation driven by impulsive noise

Considering a linear parabolic stochastic partial differential equation driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H, t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an impulsive cylindrical process and Q describes the spatial covariance structure of the noise; Tr(A^{-\alpha})0 and A^\beta Q is bounded for some \beta\in(\alpha-1,\alpha]. A discretization (X_h^n)_{n\in\{0,1,...,N\}} is defined via the finite element method in space (parameter h>0) and a \theta-method in time (parameter \Delta t=T/N). For \phi\in C^2_b(H;R) we show an integral representation for the error |E\phi(X^N_h)-E\phi(X_T)| and prove that |E\phi(X^N_h)-E\phi(X_T)|=O(h^{2\gamma}+(\Delta t)^{\gamma}) where \gamma<1-\alpha+\beta.Comment: 29 pages; Section 1 extended, new results in Appendix

On the Alekseev-Gr\"obner formula in Banach spaces

The Alekseev-Gr\"obner formula is a well known tool in numerical analysis for describing the effect that a perturbation of an ordinary differential equation (ODE) has on its solution. In this article we provide an extension of the Alekseev-Gr\"obner formula for Banach space valued ODEs under, loosely speaking, mild conditions on the perturbation of the considered ODEs.Comment: 36 page

Stochastic fiber dynamics in a spatially semi-discrete setting

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.Comment: 20 pages; typos removed, references adde