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Zbl 1088.65002
Matthies, Hermann G.; Keese, Andreas
Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. (English)
[J] Comput. Methods Appl. Mech. Eng. 194, No. 12-16, 1295-1331 (2005). ISSN 0045-7825

Authors' abstract: Stationary systems modelled by elliptic partial differential equations -- linear as well as nonlinear -- with stochastic coefficients (random fields) are considered. The mathematical setting as a variational problem, existence theorems, and possible discretisations -- in particular with respect to the stochastic part -- are given and investigated with regard to stability. Different and increasingly sophisticated computational approaches involving both Wiener's polynomial chaos as well as the Karhunen-Loève expansion are addressed in conjunction with stochastic Galerkin procedures, and stability within the Galerkin framework is established. New and effective algorithms to compute the mean and covariance of the solution are proposed. The similarities and differences with better known Monte Carlo methods are exhibited, as well as alternatives to integration in high-dimensional spaces. Hints are given regarding the numerical implementation and parallelisation. Numerical examples serve as illustration.
[Dominique Lepingle (Orléans)]

MSC 2000:: *65C30 Stochastic differential and integral equations
60H15 Stochastic partial differential equations
35R60 PDE with randomness
60H35 Computational methods for stochastic equations
35J25 Second order elliptic equations, boundary value problems
35J65 (Nonlinear) BVP for (non)linear elliptic equations
65N12 Stability and convergence of numerical methods (BVP of PDE)
65C05 Monte Carlo methods

Keywords: linear and nonlinear elliptic stochastic partial differential equations; Galerkin methods; Karhunen-Loeve expansion; Wiener's polynomial chaos; white noise analysis; sparse Smolyak quadrature; Monte Carlo methods; stochastic finite elements; parallel computation; stability; algorithms; numerical examples

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