Sparse finite elements for elliptic problems with stochastic loading

Summary.We formulate elliptic boundary value problems with stochastic loading in a bounded domain D⊂ℝd. We show well-posedness of the problem in stochastic Sobolev spaces and we derive a deterministic elliptic PDE in D×D for the spatial correlation of the random solution. We show well-posedness and regularity results for this PDE in a scale of weighted Sobolev spaces with mixed highest order derivatives. Discretization with sparse tensor products of any hierarchic finite element (FE) spaces in D yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data. Multilevel preconditioning in D×D allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation.

[1]  Joseph B. Keller,et al.  Stochastic equations and wave propagation in random media , 1964 .

[2]  Michael Griebel,et al.  Sparse grids for boundary integral equations , 1999, Numerische Mathematik.

[3]  P. Protter.  Stochastic integration and differential equations : a new approach , 1990 .

[4]  Bernt Øksendal,et al.  Stochastic differential equations (3rd ed.): an introduction with applications , 1992 .

[5]  Wolfgang Dahmen,et al.  Element-by-Element Construction of Wavelets Satisfying Stability and Moment Conditions , 1999, SIAM J. Numer. Anal..

[6]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[7]  Ivo Babuška.  On randomised solutions of Laplace's equation , 1961 .

[8]  R. Strichartz.  A Guide to Distribution Theory and Fourier Transforms , 1994 .

[9]  J. D. Fériet,et al.  Random Solutions of Partial Differential Equations , 1956 .

[10]  森山 昌彦,et al.  「確率有限要素法」(Stochastic Finite Element Method) , 1985 .

[11]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[12]  A. Matache.  Sparse two-scale FEM for homogenization problems , 2001 .

[13]  Ivo Babuška,et al.  Regularity of the solution of elliptic problems with piecewise analytic data. Part 1. Boundary value problems for linear elliptic equation of second order , 1988 .

[14]  Humberto Contreras,et al.  The stochastic finite-element method , 1980 .

[15]  A. Yaglom,et al.  An Introduction to the Theory of Stationary Random Functions , 1963 .

[16]  Bernt Øksendal,et al.  Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach , 1996 .

[17]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[18]  Michael Griebel,et al.  Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems , 1995, Adv. Comput. Math..

[19]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[20]  I. Babuska,et al.  Solution of stochastic partial differential equations using Galerkin finite element techniques , 2001 .