Inf–sup conditions for twofold saddle point problems

Abstract

Summary Necessary and sufficient conditions for existence and uniqueness of solutions are developed for twofold saddle point problems which arise in mixed formulations of problems in continuum mechanics. This work extends the classical saddle point theory to accommodate nonlinear constitutive relations and the twofold saddle structure. Application to problems in incompressible fluid mechanics employing symmetric tensor finite elements for the stress approximation is presented.

Analysis and Finite Element Approximation of a Nonlinear Stationary Stokes Problem Arising in Glaciology

“…Spaces V and Q are required to satisfy the inf-sup condition, see 5,20 , to ensure the existence and the uniqueness of p ∈ Q such that u, p satisfies the mixed formulation 3.8 . The inf-sup condition is proved in 15, 21 when Γ D ∂Ω or, equivalently, Γ N Γ R ∅ .…”

Approximation of generalized Stokes problems using dual‐mixed finite elements without enrichment

“…Thus the goal of the present analysis is to describe sufficient conditions for problems (9) to be well-posed by showing, with convenient choices of inf-sup conditions, (5) is really just a single saddle point problem (6) whose criteria for solvability has been met. This is motivated by the recent results in [9] that give equivalent sets of inf-sup conditions for twofold saddle point problems. In the following analysis, we define the P-kernel of b 2 by…”

“…Employing a result from [9], it is shown here that when using trace-free velocity gradients, it is not necessary to enrich the space to include the deviatoric of the discrete pseudostress. When combined with the elimination of the pressure from the formulation, the method described here reduces the computational cost of the method in [19] for the lowest order 2-D case by 4 degrees of freedom on each triangle (this is a true savings, as they are neither edge nor vertex degrees of freedom).…”

A Dual–Mixed Finite Element Method for the Brinkman Problem

“…Equivalent conditions for an inf-sup condition to hold over a product space were developed in [17] where it was shown that (2.7) holds and that u U ≤ C G G for pairs (u, G) ∈ Z if and only if [17,…”

“…In a manner similar to dual-mixed methods for the Stokes and Navier-Stokes problems [17,18], the work presented here constructs a dual-mixed variational formulation of (1.1) in which the fluid velocity, the fluid stress, and the deviatoric part of the velocity gradient are the primary unknowns. This construction results in a twofold saddle point problem whose coercivity is guaranteed for all meaningful α, ν under a generalized Poincaré inequality.…”