Approximation of Stochastic Partial Differential Equations

Partial Differential Equations (PDEs) play an essential role for mathematical modeling of many physical phenomena, and the literature devoted to their theory and applications is enormous. Stochastic Partial Differential Equations (SPDEs) started to appear in the mid - 1960s. They were motivated by the need to describe random phenomena studied in the natural sciences such as control theory, physics, chemistry and biology. As in the theory of PDEs, often only existence and uniqueness can be shown, but theie are very few SPDEs for which analytical solutions can be obtained, and properties of the solution cannot be found by direct calculation. Here, theory and numerical work often go hand in hand: pictures obtained numerically can lead to conjectures. These conjectures can be verified by theory - or an the other hand: Verifying conjectures by numerical experiments is much quicker than verifying a conjecture by theory. In contrast to the theory of PDEs, only some scattered works exist about the numerical simulation of SPDEs. Also due to the peculiarities of the stochastic perturbation, such as nowhere differentiability and infmite variation, the methods which work in the deterministie Gase usually cannot be transferred to the stochastic Gase. In the project we will consider the Numerical Approximation of Parabolic SPDEos. Here, the main emphasis will be an nonlinear SPDEs and an SPDEs driven by Poisson random measure. In physics the most fundamental equations are nonlinear. These nonlinearities yield to new phenomenas, which orte cannot sec in the linear case. Thus, my first point of investigation will lie an SPDEs with unbounded nonlinear perturbations, such as the stochastic Navier-Stokes equation, the stochastic Burgers Equation, or the reaction- diffusion equations with polynomial nonlinearities. Stochastic Navier-Stokes equations have recently been paid a considerable attention in physical literature in connection with the study of turbulence. The Burgers equation appears in a number of physical Problems, for instance, the formation of large clusters in the universe, or the kinetic roughening of growing surfaces. Stochastic reaction-diffusion equations appear e.g. as models for chemical autocatalytic reactions or in population dynamics. The second point of investigations will be SPDEs driven by Poisson random measure. For instance - in neurophysiology the driving noise of the cable equation is basically impulsive, e.g. of Poisson type. Thus, from the point of view of applications; to handle such cases orte can replace the Gaussian noise by a Poisson random measure.