Fine error bounds for approximate asymmetric saddle point problems

TL;DR

This paper establishes refined global error bounds for asymmetric saddle point problems in mixed finite element methods, clarifying the role of inf-sup constants and improving understanding of stability and approximation quality.

Contribution

It introduces finer error bounds for asymmetric saddle point problems, explicitly relating stability constants to approximation accuracy, and clarifies the conditions for well-posedness.

Findings

Derived global error bounds depending on inf-sup constants

Identified the three key constants for stability analysis

Provided explicit expressions for the stability constant

Abstract

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. This topic was mostly studied for variational formulations defined upon the same product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever either these spaces or the two bilinear forms involving the multiplier are distinct, the saddle point problem is asymmetric. The three inf-sup conditions to be satisfied by the product spaces stipulated in work on the subject, in order to guarantee well-posedness, are well known. However, the material encountered in the literature addressing the approximation of this class of problems left room for improvement and clarifications. After making a brief review of the existing contributions to the topic that justifies such an assertion, in this paper we set up finer global error bounds for the pair primal variable-multiplier solving an asymmetric saddle point problem. Besides well-posedness, the three constants in the aforementioned inf-sup conditions are identified as all that is needed for determining the stability constant appearing therein, whose expression is exhibited. As a complement, refined error bounds depending only on these three constants are given for both unknowns separately.