A finite element method for elliptic problems with stochastic input data

We compute the expectation and the two-point correlation of the solution to elliptic boundary value problems with stochastic loadings. In case of elliptic problems on stochastic domains or with stochastic coefficients analogous expressions hold to leading order in the size of the stochastic perturbation. The solution's two-point correlation satisfies a deterministic tensor product partial differential equation on the twofold product domain. For its numerical solution we apply a sparse tensor product approximation by multilevel frames. This way standard finite element techniques can be used. Numerical examples illustrate feasibility and scope of the method.

[1]  Daniel M. Tartakovsky,et al.  Numerical Methods for Differential Equations in Random Domains , 2006, SIAM J. Sci. Comput..

[2]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[3]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[4]  Christoph Schwab,et al.  Sparse finite element methods for operator equations with stochastic data , 2006 .

[5]  E. F. Kaasschieter,et al.  Preconditioned conjugate gradients for solving singular systems , 1988 .

[6]  Karsten Eppler,et al.  Boundary integral representations of second derivatives in shape optimization , 2000 .

[7]  Hans-Joachim Bungartz,et al.  Acta Numerica 2004: Sparse grids , 2004 .

[8]  M. Fornasier,et al.  Adaptive Frame Methods for Elliptic Operator Equations: The Steepest Descent Approach , 2007 .

[9]  Reinhold Schneider,et al.  Multilevel frames for sparse tensor product spaces , 2008, Numerische Mathematik.

[10]  Christoph Schwab,et al.  Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients , 2007 .

[11]  Reinhold Schneider,et al.  Sparse second moment analysis for elliptic problems in stochastic domains , 2008, Numerische Mathematik.

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

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

[14]  Wolfgang Alt,et al.  Rheinisch-westfälische Technische Hochschule Aachen Energetics and Dynamics of Global Integrals Modeling Interaction between Stiff Filaments Energetics and Dynamics of Global Integrals Modeling Interaction between Stiff Filaments , 2022 .

[15]  Helmut Harbrecht,et al.  On output functionals of boundary value problems on stochastic domains , 2009 .

[16]  Karsten Eppler,et al.  Optimal Shape Design for Elliptic Equations Via Bie-Methods , 2000 .

[17]  Fabio Nobile,et al.  Worst case scenario analysis for elliptic problems with uncertainty , 2005, Numerische Mathematik.

[18]  BabuskaIvo,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007 .

[19]  Rob P. Stevenson,et al.  Adaptive Solution of Operator Equations Using Wavelet Frames , 2003, SIAM J. Numer. Anal..

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

[21]  Gabriel Wittum,et al.  Multigrid Methods V , 1998 .

[22]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[23]  H. Bungartz,et al.  Sparse grids , 2004, Acta Numerica.

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

[25]  Peter Oswald,et al.  On Function Spaces Related to Finite Element Approximation Theory , 1990 .

[26]  Michael Griebel,et al.  Optimized general sparse grid approximation spaces for operator equations , 2009, Math. Comput..

[27]  Claudio Canuto,et al.  A fictitious domain approach to the numerical solution of PDEs in stochastic domains , 2007, Numerische Mathematik.

[28]  Hans-Joachim Bungartz,et al.  A multigrid algorithm for higher order finite elements on sparse grids. , 1997 .

[29]  William Rundell,et al.  A Second Degree Method for Nonlinear Inverse Problems , 1999, SIAM J. Numer. Anal..

[30]  H. Bungartz,et al.  Sparse Grids: Recent Developments for Elliptic Partial Differential Equations , 1998 .

[31]  P. Oswald,et al.  On additive Schwarz preconditioners for sparse grid discretizations , 1993 .

[32]  Olaf Steinbach,et al.  Numerische Näherungsverfahren für elliptische Randwertprobleme , 2003 .

[33]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[34]  Sergio Albeverio,et al.  Factorisation of Non-negative Fredholm Operators and Inverse Spectral Problems for Bessel Operators , 2009 .

[35]  Alexander Pfaffinger,et al.  Funktionale Beschreibung und Parallelisierung von Algorithmen auf dünnen Gittern , 1997 .

[36]  Christoph Zenger,et al.  The Solution of Multidimensional Real Helmholtz Equations on Sparse Grids , 1996, SIAM J. Sci. Comput..

[37]  Hermann G. Matthies,et al.  Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations , 2005 .

[38]  Hanno Gottschalk,et al.  A Comment on the Infra-Red Problem in the AdS/CFT Correspondence , 2007 .

[39]  Habib N. Najm,et al.  Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes , 2005, SIAM J. Sci. Comput..

[40]  James H. Bramble,et al.  The Lagrange multiplier method for Dirichlet’s problem , 1981 .

[41]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[42]  Christoph Schwab,et al.  Sparse finite elements for elliptic problems with stochastic loading , 2003, Numerische Mathematik.

[43]  J. Pasciak,et al.  Parallel multilevel preconditioners , 1990 .

[44]  Michael Griebel,et al.  Multilevelmethoden als Iterationsverfahren über Erzeugendensystemen , 1994 .

[45]  Michael Griebel,et al.  Multilevel Algorithms Considered as Iterative Methods on Semidefinite Systems , 1994, SIAM J. Sci. Comput..

[46]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[47]  Long Chen,et al.  Stability and accuracy of adapted finite element methods for singularly perturbed problems , 2008, Numerische Mathematik.

[48]  Christoph Schwab,et al.  Sparse Finite Elements for Stochastic Elliptic Problems – Higher Order Moments , 2003, Computing.

[49]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[50]  Peter Oswald,et al.  Multilevel Finite Element Approximation , 1994 .

[51]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..