A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes
A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes
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DOI：
10.3929/ethz-a-010386302
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发表时间：
2013-12
期刊：
影响因子：
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通讯作者：
M. Eigel;C. J. Gittelson;C. Schwab;E. Zander
中科院分区：
文献类型：
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作者：
M. Eigel;C. J. Gittelson;C. Schwab;E. Zander
我们分析了后验误差估计和自适应加密算法的随机有限元方法的可数参数，椭圆边值问题。本文建立了一个残差估计器，它在能量范数下分离了参数空间中gpc-Galerkin离散和物理空间中有限元离散的影响。证明了自适应算法的收敛性。为此，它的迭代的收缩性质被证明。结果表明，该算法在有限元离散有源gpc系数时产生的三角剖分序列是渐近最优的。数值实验验证了理论结果。
We analyze a posteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods for countably-parametric, elliptic boundary value problems. A resid- ual error estimator which separates the effects of gpc-Galerkin discretization in parameter space and of the Finite Element discretization in physical space in energy norm is established. It is proved that the adaptive algorithm converges. To this end, a contraction property of its iterates is proved. It is shown that the sequences of triangulations which are produced by the algorithm in the FE discretization of the active gpc coefficients are asymptotically optimal. Numerical experiments illustrate the theoretical results.