The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle-point formulation is well established since many decades. However, this topic was mostly studied for variational formulations defined upon the same finite-element product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever these two product spaces are different the saddle point problem is asymmetric. It turns out that the conditions to be satisfied by the finite elements product spaces stipulated in the few works on this case may be of limited use in practice. The purpose of this paper is to provide an in-depth analysis of the well-posedness and the uniform stability of asymmetric approximate saddle point problems, based on the theory of continuous linear operators on Hilbert spaces. Our approach leads to necessary and sufficient conditions for such properties to hold, expressed in a readily exploitable form with fine constants. In particular standard interpolation theory suffices to estimate the error of a conforming method.

翻译：混合有限元方法用于求解鞍点形式下各类椭圆型偏微分方程的理论已建立数十年。然而，该课题主要针对定义在相同有限元乘积空间（即主变量-乘子的试函数对与检验函数对）上的变分形式进行研究。当这两个乘积空间不同时，鞍点问题表现为非对称性。现有少数文献对此情形提出的有限元乘积空间需满足的条件，在实际应用中可能具有局限性。本文旨在基于希尔伯特空间上的连续线性算子理论，对非对称近似鞍点问题的适定性与一致稳定性进行深入分析。我们的方法给出了此类性质成立的充分必要条件，并以易于利用的形式呈现了精细常数。特别地，标准插值理论即可用于估计协调方法的误差。