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Sparse finite element methods for operator equations with stochastic data

Tobias von Petersdorff, Christoph Schwab (2006)

Applications of Mathematics

Let A V V ' be a strongly elliptic operator on a d -dimensional manifold D (polyhedra or boundaries of polyhedra are also allowed). An operator equation A u = f with stochastic data f is considered. The goal of the computation is the mean field and higher moments 1 u V , 2 u V V , ... , k u V V of the solution. We discretize the mean field problem using a FEM with hierarchical basis and N degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment k u for k 1 . The key tool...

Spectral discretization of Darcy equations coupled with Navier-Stokes equations by vorticity-velocity-pressure formulation

Yassine Mabrouki, Jamil Satouri (2022)

Applications of Mathematics

We consider a model coupling the Darcy equations in a porous medium with the Navier-Stokes equations in the cracks, for which the coupling is provided by the pressure's continuity on the interface. We discretize the coupled problem by the spectral element method combined with a nonoverlapping domain decomposition method. We prove the existence of solution for the discrete problem and establish an error estimation. We conclude with some numerical tests confirming the results of our analysis.

Spectral methods for singular perturbation problems

Wilhelm Heinrichs (1994)

Applications of Mathematics

We study spectral discretizations for singular perturbation problems. A special technique of stabilization for the spectral method is proposed. Boundary layer problems are accurately solved by a domain decomposition method. An effective iterative method for the solution of spectral systems is proposed. Suitable components for a multigrid method are presented.

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