The increasing demand on the quality and reliability of numerical simulations of physical problems results in an increasing complexity of mathematical models. Especially, the knowledge for the description and definition of model relevant parameters often cannot be assumed to be available in a deterministic way. There are often uncertainties involved, which can arise for example by inexact measurements or modelling assumptions. This makes the development of appropriate and efficient numerical solution methods a crucial task.
The focus of this project is the development of parallel numerical methods for uncertainty propagation in partial differential equations (PDEs) using Polynomial Chaos and stochastic Galerkin projection. The spatial components of the PDEs considered are discretized by Finite-Elements giving rise to the Spectral-Stochastic-Finite-Element-Method (SSFEM). PDEs ranging from linear elliptic type to the incompressible Navier-Stokes equations in steady and unsteady formulation serve as applications based on probability models for the corresponding uncertain parameters. Most attention is devoted to the development of parallel<span”> numerical methods for the efficient solution of the associated discretized systems. Parallelization is carried out in the stochastic domain, the spatial domain and temporal domain using distributed and shared memory approaches.
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