Fine error bounds for approximate asymmetric saddle point problems

• Published in: Computational & applied mathematics Vol. 43; no. 4
• Main Author: Ruas, Vitoriano
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Asymmetry; Computational Mathematics and Numerical Analysis; Constants; Elliptic functions; Errors; Finite element method; Mathematical analysis; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Multipliers; Partial differential equations; Saddle points; Well posed problems; 70G75; 74A15; Error bounds; 78M30; Inf-sup conditions; Stability constants; 76M30; Asymmetric saddle-point problem; 65N30; 80M30; Mixed finite elements
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-024-02678-7

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, to guarantee well-posedness, are known [(see, e.g., Exercise 2.14 of Ern and Guermond (Theory and practice of finite elements, Applied mathematical, sciences, Springer, 2004)]. 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.