稳定Brinkman-Forchheimer方程与双扩散方程耦合的Banach空间全混合公式

IF 1.9 3区 数学 Q2 Mathematics
Sergio Caucao, G. Gatica, J. Ortega
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引用次数: 10

摘要

针对稳态Brinkman—Forchheimer方程与双扩散方程耦合所引起的非线性问题,提出并分析了一种新的混合有限元方法。除了速度、温度和浓度之外,我们的方法还引入了速度梯度、伪应力张量和一对涉及温度/浓度、其梯度和速度的向量,作为进一步的未知数。因此,我们得到了一个完全混合变分公式,在每组方程中表示一个巴拿赫空间框架。通过这种方式,与以前为这个和相关耦合问题开发的技术不同,现在不需要将增广过程纳入公式或可解性分析中。然后将得到的非增广格式等效化为不动点方程,从而利用著名的Banach定理,结合Banach空间中非线性单调算子的经典结果和babuska - brezzi理论,证明了连续系统和离散系统的唯一可解性。给出了满足离散自适应条件的适当有限元子空间,并推导了最优先验误差估计。几个数值算例证实了理论的收敛速度,并说明了该方法的性能和灵活性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman-Forchheimer and double-diffusion equations
We propose and analyze a new mixed finite element method for the nonlinear problem given by the coupling of the steady Brinkman--Forchheimer and double-diffusion equations. Besides the velocity, temperature, and concentration, our approach introduces the velocity gradient, the pseudostress tensor, and a pair of vectors involving the temperature/concentration, its gradient and the velocity, as further unknowns. As a consequence, we obtain a fully mixed variational formulation presenting a Banach spaces framework in each set of equations. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and the Babusska-Brezzi theory in Banach spaces, are applied to prove the unique solvability of the continuous and discrete systems. Appropriate finite element subspaces satisfying the required discrete inf-sup conditions are specified, and optimal a priori error estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method.
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来源期刊
CiteScore
2.70
自引率
5.30%
发文量
27
审稿时长
6-12 weeks
期刊介绍: M2AN publishes original research papers of high scientific quality in two areas: Mathematical Modelling, and Numerical Analysis. Mathematical Modelling comprises the development and study of a mathematical formulation of a problem. Numerical Analysis comprises the formulation and study of a numerical approximation or solution approach to a mathematically formulated problem. Papers should be of interest to researchers and practitioners that value both rigorous theoretical analysis and solid evidence of computational relevance.
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