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The theory of mixed finite element methods for solving different types ofelliptic partial differential equations in saddle-point formulation is wellestablished since many decades. However, this topic was mostly studied forvariational formulations defined upon the same finite-element product spaces ofboth shape- and test-pairs of primal variable-multiplier. Whenever these twoproduct spaces are different the saddle point problem is asymmetric. It turnsout that the conditions to be satisfied by the finite elements product spacesstipulated in the few works on this case may be of limited use in practice. Thepurpose of this paper is to provide an in-depth analysis of the well-posednessand the uniform stability of asymmetric approximate saddle point problems,based on the theory of continuous linear operators on Hilbert spaces. Ourapproach leads to necessary and sufficient conditions for such properties tohold, expressed in a readily exploitable form with fine constants. Inparticular standard interpolation theory suffices to estimate the error of aconforming method.

Fine stability constants and error bounds for asymmetric approximate saddle point problems