ADAPTIVE MULTILEVEL METHODS IN 3 SPACE DIMENSIONS
ADAPTIVE MULTILEVEL METHODS IN 3 SPACE DIMENSIONS
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DOI：
10.1002/nme.1620361808
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发表时间：
1993-09-30
影响因子：
2.9
通讯作者：
KORNHUBER, R
中科院分区：
文献类型：
--
作者：
BORNEMANN, F;ERDMANN, B;KORNHUBER, R
本文考虑三维空间自伴椭圆问题的近似解，采用分段线性有限元方法，在高度非均匀的四面体网格上自适应生成。提出的线性方程组用共轭梯度法迭代求解，并给出了多级预条件子。这里，迭代解的精度与离散化误差相耦合。由于层次基预处理器的性能在三维空间中恶化，因此使用BPX预处理器，特别注意有效的实现。可靠的后验估计的离散误差来自局部比较与分段二次元素的近似结果。为了说明的理论结果，我们考虑一个熟悉的模型问题，涉及凹角和现实生活中的问题所产生的热疗，最近的临床癌症治疗方法。
We consider the approximate solution of self-adjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non-uniform tetrahedral mesh which is generated adaptively. The arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner. Here, the accuracy of the iterative solution is coupled with the discretization error. As the performance of hierarchical bases preconditioners deteriorates in three space dimensions, the BPX preconditioner is used, taking special care of an efficient implementation. Reliable a posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements. To illustrate the theoretical results, we consider a familiar model problem involving reentrant corners and a real-life problem arising from hyperthermia, a recent clinical method for cancer therapy.