Volume 5, Issue 2-4
Error Control in Multi-Element Generalized Polynomial Chaos Method for Elliptic Problems with Random Coefficients

Xiaoliang Wan & George Em Karniadakis

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

Published online: 2009-02

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  • Abstract

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 ≤ 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.

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@Article{CiCP-5-793, author = {}, title = {Error Control in Multi-Element Generalized Polynomial Chaos Method for Elliptic Problems with Random Coefficients}, journal = {Communications in Computational Physics}, year = {2009}, volume = {5}, number = {2-4}, pages = {793--820}, abstract = {

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 ≤ 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.

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7764.html} }
TY - JOUR T1 - Error Control in Multi-Element Generalized Polynomial Chaos Method for Elliptic Problems with Random Coefficients JO - Communications in Computational Physics VL - 2-4 SP - 793 EP - 820 PY - 2009 DA - 2009/02 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cicp/7764.html KW - AB -

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 ≤ 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.

Xiaoliang Wan & George Em Karniadakis. (2020). Error Control in Multi-Element Generalized Polynomial Chaos Method for Elliptic Problems with Random Coefficients. Communications in Computational Physics. 5 (2-4). 793-820. doi:
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