A hybrid two-step finite element method for flux approximation: a priori estimates
A hybrid two-step finite element method for flux approximation: a priori estimates
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通量近似的混合两步有限元方法：先验估计
DOI：
10.1051/m2an/2016062
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
D. Sheen
中科院分区：
文献类型：
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作者：
Jaeun Ku;Young Ju Lee;D. Sheen
在传统的混合有限元方法的基础上，提出了一种新的基于网格尺寸杂交的两步法。在粗网格上，主变量用标准的Galerkin方法近似，其计算量很小。然后，在精细网格上，寻求对偶变量的H（div）投影作为通量变量的精确近似。我们的方法不依赖于传统的混合列式框架，因此，对有限元空间的选择是免费的inf-sup稳定性条件的要求。更确切地说，我们的方法是制定在一个完全解耦的方式，仍然实现了最佳的误差收敛顺序。这导致了一个计算策略更容易和更广泛的实现比混合有限元方法。此外，独立提出的解决方案的策略，允许使用不同的网格以及不同的离散化方案在计算的主要和通量变量。我们证明了细网格尺寸h可以取为粗网格尺寸H的平方，或者取为具有适当选择的参数δ的高阶幂。这意味着粗网格解决方案的计算成本与细网格解决方案相比可以忽略不计。事实上，数值实验表明，使用我们的策略相比，混合有限元方法的优势。文中还给出了选择最佳参数δ的一些准则。此外，我们的方法是提供一个渐近精确的后验误差估计的主要变量p在H1范数。
We present a new two–step method based on the hybridization of mesh sizes in the traditional mixed finite element method. On a coarse mesh, the primary variable is approximated by a standard Galerkin method, whose computational cost is very low. Then, on a fine mesh, an H (div) projection of the dual variable is sought as an accurate approximation for the flux variable. Our method does not rely on the framework of traditional mixed formulations, the choice of pair of finite element spaces is, therefore, free from the requirement of inf-sup stability condition. More precisely, our method is formulated in a fully decoupled manner, still achieving an optimal error convergence order. This leads to a computational strategy much easier and wider to implement than the mixed finite element method. Additionally, the independently posed solution strategy allows to use different meshes as well as different discretization schemes in the calculation of the primary and flux variables. We show that the finer mesh size h can be taken as the square of the coarse mesh size H , or a higher order power with a proper choice of parameter δ . This means that the computational cost for the coarse-grid solution is negligible compared to that for the fine-grid solution. In fact, numerical experiments show an advantage of using our strategy compared to the mixed finite element method. Some guidelines to choose an optimal parameter δ are also given. In addition, our approach is shown to provide an asymptotically exact a posteriori error estimator for the primary variable p in H 1 norm.