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Coons-patch macroelements in two-dimensional parabolic problems

机译：二维抛物线问题中的Coons-patch宏元素

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摘要

Having recently obtained encouraging results in elliptic and hyperbolic problems, this paper summarizes previous work and further investigates the performance of large isoparametric finite elements based on the Coons-Gordon interpolation formula in the analysis of two-dimensional parabolic potential problems. The latter formula allows the global interpolation of the potential within the whole problem domain and leads to the so-called Coons-patch-macroelements (CPM), where the degrees of freedom appear primarily at the element boundaries but in the general case it is also possible to use any desirable number of internal nodes. Mathematical and numerical aspects such as the relationship between boundary-only Coons-patch macroelements and Serendipity type elements, the systematic and straightforward way of adding internal nodes, the procedure of merging dissimilar domains and, finally, efficient numerical integration schemes are discussed. Numerical results on typical static (Laplace) and time-dependent thermal problems sustain the proposed method, which is successfully compared with conventional bilinear finite elements and exact analytical solutions.
机译：最近在椭圆和双曲问题上获得了令人鼓舞的结果，本文总结了以前的工作，并进一步研究了基于Coons-Gordon插值公式的大型等参有限元在二维抛物线势问题分析中的性能。后一个公式允许对整个问题域内的势进行全局插值，并导致所谓的Coons-patch-macroelements（CPM），其中自由度主要出现在元素边界，但通常情况下也是可以使用任意数量的内部节点。数学和数值方面，例如仅边界的Coons-patch宏元素与Serendipity类型元素之间的关系，添加内部节点的系统而直接的方法，合并不同域的过程以及最后的有效数值积分方案。典型的静态（拉普拉斯）和时变热问题的数值结果支持了该方法，该方法已成功与常规双线性有限元和精确的解析解进行了比较。

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