Sparse finite elements for elliptic problems with stochastic loading
Sparse finite elements for elliptic problems with stochastic loading
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DOI：
10.1007/s00211-003-0455-z
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发表时间：
2003-10
影响因子：
2.1
通讯作者：
C. Schwab;R. Todor
中科院分区：
文献类型：
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作者：
C. Schwab;R. Todor
在有界域D⊂ℝd上建立了具有随机载荷的椭圆型边值问题，证明了该问题在随机Sobolov空间中的适定性，并给出了随机解的空间相关性的确定性椭圆型偏微分方程解。我们在具有混合最高阶导数的加权Soblev空间的尺度上证明了这类偏微分方程解的适定性和正则性。用稀疏张量积离散空间中的任何分层有限元(FE)空间，即使在存在奇点或空间完全不相关的情况下，也可以获得空间相关的最优渐近收敛速度。多级预条件IND×D允许迭代求解相关核的离散方程，其复杂性基本上与平均场方程的解相同。
We formulate elliptic boundary value problems with stochastic loading in a bounded domainD⊂ℝd. We show well-posedness of the problem in stochastic Sobolev spaces and we derive a deterministic elliptic PDE inD×Dfor 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 inDyields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data. Multilevel preconditioning inD×Dallows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation.