A Survey of Mixed Finite Element Methods

Abstract

IThis paper is an introduction to and an overview of mixed finite element methods. It discusses the mixed formulation of certain basic problems in elasticity and hydrodynamics. It also discusses special techniques for solving the discrete problem.

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

“…shared by elements K and L. For a general reference to mixed finite element methods, we refer to Brezzi and Fortin [20] or Roberts and Thomas [56].…”

“…We always give a detailed form of the estimates up to the form with the error between the exact solution and its interpolate. Obtaining the final error estimates is then a question of application of interpolation estimates, presented, e.g., in [20,52,56]. For the sake of completeness, we include these final results, supposing the full necessary regularity; here C denotes a generic constant independent of h. …”

“…Indeed, it follows by subtracting (1.4a) from (1.5a) and using (2.3) that Noting that ∇ · u h = P Φ h (f ) by (1.5b) and using the interpolation estimates, see, e.g., [20,52,56] we infer from the previous results the following corollary: 4a)-(1.4b), u h , p h by (1.5a)-(1.5b), andp h by (4.2a)-(4.2b). Then…”

“…Here Φ h ⊂ L 2 (Ω) and V h ⊂ H(div, Ω) are some of the usual finite-dimensional spaces defined on a mesh T h of simplices or rectangular parallelepipeds; see Section 4.1 below and Brezzi and Fortin [20] or Roberts and Thomas [56]. The main purposes of this paper are the following: i) present a unified framework for both a priori and a posteriori error analysis of mixed finite element methods; ii) base this framework entirely on the primal weak formulation (1.2) (and its above-cited direct equivalence with (1.4a)-(1.4b)) on the continuous level and on postprocessing and the discrete Friedrichs inequality on the discrete level; in particular, the explicit use of the uniform-in-h discrete inf-sup condition is avoided; iii) arrive at optimal a priori estimates (under minimal necessary assumptions); iv) present new (and optimal) a posteriori error estimates; v) obtain these results with as simple as possible proofs; vi) present some new (to the best of the author's knowledge) properties of the mixed finite element methods; vii) give a general framework for a posteriori error estimation in locally conservative methods.…”

“…Finally, by construction, p h is the L 2 (Ω)-orthogonal projection ofp h onto Φ h , so that the estimates for the error in p − p h are easily recovered. The analysis still relies on the appropriate vector interpolation operator of each mixed finite element method, satisfying the commuting diagram property; see [20,Section III.3]. On the other hand, the use of the uniform-in-h discrete inf-sup condition is avoided by the postprocessing and the discrete Friedrichs inequality; for some related comments on this last point, we refer to [30,31] and [8,Theorem 2].…”