abstract:
We present an abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive Lévy noise. We first derive a representation formula for the error which we then apply to study space-time discretizations of the stochastic heat and wave equations. We use the standard discontinuous finite element method as spatial discretization and the backward Euler method respectively I-stable rational approximations to the exponential function as time-stepping for the heat and wave equations. For twice continuously differentiable test functions with bounded first and second derivatives, with some extra condition on the second derivative for the wave equation, the weak rate is found to be twice that of the strong rate. The results extend the earlier work by the Lindner and Schilling as we consider general square-integrable infinite dimensional Lévy processes with no additional assumptions on the jump intensity measure. Furthermore, the present framework is applicable to hyperbolic equations as well.
This is a joint work with Felix Lindner (TU Kaiserslautern) and Renee Schilling (TU Dresden). |