TY - JOUR
T1 - Error Control in Multi-Element Generalized Polynomial Chaos Method for Elliptic Problems with Random Coefficients
AU - Xiaoliang Wan & George Em Karniadakis
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 - <p style="text-align: justify;">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 (<em>O</em>(10)) of random dimensions. We employ low-order (<em>p</em> ≤ 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 <em>hp</em>-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.</p>