Error Control in Finite Element Computations
Error Control in Finite Element Computations
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有限元计算中的误差控制
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
R. Rannacher
中科院分区：
文献类型：
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作者：
R. Rannacher
我们给出了有限元Galerkin方法中后验误差控制和自适应网格设计的一般范例。在有限元方法中，传统的误差控制策略是基于包含计算解的局部残差的整体能量或L 2范数误差的后验估计。这种估计包含描述有限元空间的局部逼近性质和线性化对偶问题的稳定性性质的常数。然后，网格细化的目标是局部误差指标的平衡。然而，通过控制全局范数中的误差而生成的网格可能不适合于局部误差量，如点值或线积分，以及在系数强烈变化的情况下。这一缺陷可以通过在后验误差估计中引入某些权重因子来克服，该加权因子依赖于对偶解并且包含有关相关误差传播的信息。这样，可以为各种误差测量生成最经济的网格。首先对一个简单的模型情况进行了系统的发展，然后用流体力学、弹塑性和辐射传递中更复杂的问题的结果来说明这一点。
We present a general paradigm for a posteriori error control and adaptive mesh design in finite element Galerkin methods. The conventional strategy for controlling the error in finite element methods is based on a posteriori estimates for the error in the global energy or L 2-norm involving local residuals of the computed solution. Such estimates contain constants describing the local approximation properties of the finite element spaces and the stability properties of a linearized dual problem. The mesh refinement then aims at the equilibration of the local error indicators. However, meshes generated via controlling the error in a global norm may not be appropriate for local error quantities like point values or line integrals and in case of strongly varying coefficients. This deficiency may be overcome by introducing certain weight-factors in the a posteriori error estimates which depend on the dual solution and contain information about the relevant error propagation. This way, optimally economical meshes may be generated for various kinds of error measures. This is systematically developed first for a simple model case and then illustrated by results for more complex problems in fluid mechanics, elasto-plasticity and radiative transfer.