Commun. Comput. Phys., 5 (2009), pp. 793-820.

Error Control in Multi-Element Generalized Polynomial Chaos Method for Elliptic Problems with Random Coefficients

Xiaoliang Wan 1*, George Em Karniadakis 1

1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.

Received 23 August 2007; Accepted (in revised version) 25 January 2008
Available online 1 August 2008


We develop the theory for a robust and efficient adaptive multi-element generalized polynomial chaos (ME-gPC) method for elliptic equations with random coefficients for a moderate number (O(10)) of random dimensions. We employ low-order ($p\leq 3$) polynomial chaos and refine the solution using adaptivity in the parametric space. We first study the approximation error of ME-gPC and prove its hp-convergence. We subsequently generate local and global a posteriori error estimators. In order to resolve the error equations efficiently, we construct a reduced space using much smaller number of terms in the enhanced polynomial chaos space to capture the errors of ME-gPC approximation. Based on the a posteriori estimators, we propose and implement an adaptive ME-gPC algorithm for elliptic problems with random coefficients. Numerical results for convergence and efficiency are also presented.

AMS subject classifications: 65C20, 65C30

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Key words: Stochastic PDE, a posteriori error estimate, elliptic problems, adaptive numerical methods, uncertainty quantification.

*Corresponding author.
Email: (X. Wan), (G. E. Karniadakis)

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