NDLI: Sparse Tensor Discretization of Elliptic sPDEs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Bieri, Marcel Schwab, Christoph Andreev, Roman
Copyright Year	2009
Abstract	We propose and analyze sparse deterministic-stochastic tensor Galerkin finite element methods (sparse sGFEMs) for the numerical solution of elliptic partial differential equations (PDEs) with random coefficients in a physical domain $D\subset\mathbb{R}^d$. In tensor product sGFEMs, the variational solution to the boundary value problem is approximated in tensor product finite element spaces $V^\Gamma\otimes V^D$, where $V^\Gamma$ and $V^D$ denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces, respectively. These approaches lead to sGFEM algorithms of complexity $O(N_\Gamma N_D)$, where $N_\Gamma=\dim V^\Gamma$ and $N_D=\dim V^D$. In this work, we use hierarchic sequences $V^\Gamma_1\subset V^\Gamma_2\subset\ldots$ and $V^D_1\subset V^D_2\subset\ldots$ of finite dimensional spaces to approximate the law of the random solution. The hierarchies of approximation spaces allow us to define sparse tensor product spaces $V^\Gamma_\ell\hat{\otimes}V^D_\ell$, $\ell=1,2,\dots$, yielding algorithms of $O(N_\Gamma\log N_D+N_D\log N_\Gamma)$ work and memory. We estimate the convergence rate of sGFEM for algebraic decay of the input random field KarhunenLove coefficients. We give an algorithm for an input adapted a-priori selection of deterministic and stochastic discretization spaces. The convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to Monte Carlo approximations. Numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach.
Starting Page	4281
Ending Page	4304
Page Count	24
File Format	PDF
ISSN	10648275
DOI	10.1137/090749256
e-ISSN	10957197
Issue Number	6
Volume Number	31
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-12-04
Access Restriction	Subscribed
Subject Keyword	Stability and convergence of numerical methods uncertainty quantification Models, numerical methods Stochastic partial differential equations stochastic partial differential equations multilevel approximations Error bounds stochastic finite element methods sparse tensor products Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods Partial differential equations with randomness, stochastic partial differential equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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