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1 in connection with the Kelvin principle leads to a symmetric, weakly coercive mixed-Galerkin formulation for the equations of linear elasticity. 4 Examples 27 and the minimization problem min v∈[H01 (Ω )]d JS (v; f) ∇ · v = 0 in Ω . 58) is a constrained optimization problem. Lagrange multiplier method. 58) by introducing a Lagrange multiplier p ∈ L02 (Ω ) and the Lagrangian functional LS (v, p; f) = JS (v; f) − p∇ · v dΩ . 58) is transformed into the unconstrained optimization problem of finding the saddle points {v, p} ∈ [H01 (Ω )]d × L02 (Ω ) of LS (v, p; f).

34) but make the resulting algebraic systems easier to solve are referred to as enhanced methods. 9) as an abstract setting for a brief overview of the many modifications that have been proposed. 9). 9) seek to make the discrete systems “easier” to solve. In come cases, they also result in a reduction of the Tikhonov regularization for which the original problem is perturbed into a problem that is wellposed and/or easier to solve. , [337]), a process that involves the construction of an externally defined problem.

15) that allow for the elimination of ph and for the replacement of the indefinite saddle-point matrix by a positive definite matrix. 10). 15). 34). 10) is defined by a single discrete space, it seems reasonable to assume that that problem is well-posed as long as V h ⊂ V . Unfortunately, this is not true. 10) can be converted to T Ah Bh vh Bh −εIh ph = dh 0 . 10) remains associated with a saddle-point formulation in which the space Sh is implicitly defined from V h by Sh = BhV h ; see [273]. 10) is contingent upon the compatibility of V h and the implicitly defined space Sh .