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Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

机译：伪无残留气泡在对流扩散反应问题稳定化中的应用

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摘要

It is known that the enrichment of the polynomial finite element space of degree 1 by bubble functions results in a stabilized scheme of the SUPG-type for the convection-diffusion-reaction problems. In particular, the residual-free bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless the configuration is simple. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. Here we propose a stabilization technique based on the RFB method and particularly designed to treat the most interesting case of small diffusion. We replace the RFB functions by their cheap, yet efficient approximations which retain the same qualitative behavior. The approximate bubbles are computed on a suitable sub-grid, the choice of whose nodes are critical and determined by minimizing the residual of a local problem with respect to L_1 norm. The resulting numerical method has similar stability features with the RFB method for the whole range of problem parameters. This fact is also confirmed by numerical experiments. We also note that the location of the sub-grid nodes suggested by the strategy herein coincides with the one in Brezzi et al. (Math. Models Methods Appl. Sci. 13:445-461, 2003).
机译：众所周知，气泡对函数的次数为1的多项式有限元空间的富集导致SUPG型对流扩散反应问题的稳定方案。特别是，无残留气泡（RFB）可以确保方法稳定，但除非配置简单，否则通常很难计算。因此，重要的是设计一种数值算法，为RFB函数提供廉价的近似值，从而为整个数值方法提供良好的稳定效果。在这里，我们提出一种基于RFB方法的稳定技术，该稳定技术专门用于处理最有趣的小扩散情况。我们用便宜却有效的近似值替换了RFB函数，它们保留了相同的定性行为。近似气泡在合适的子网格上计算，其子节点的选择很关键，并通过最小化关于L_1范数的局部问题的残差来确定。对于整个问题参数范围，所得的数值方法都具有与RFB方法类似的稳定性特征。数值实验也证实了这一事实。我们还注意到，这里的策略建议的子网格节点的位置与Brezzi等人的一致。 （Math.Models Methods Appl.Sci.13：445-461，2003）。

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