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A minimal stabilisation procedure for mixed finite element methods. (English) Zbl 1009.65067
Summary: Stabilization methods are often used to circumvent the difficulties associated with the stability of mixed finite element methods. Stabilization however also means an excessive amount of dissipation or the loss of nice conservation properties. It would thus be desirable to reduce these disadvantages to a minimum. We present a general framework, not restricted to mixed methods, that permits to introduce a minimal stabilizing term and hence a minimal perturbation with respect to the original problem.
To do so, we rely on the fact that some part of the problem is stable and should not be modified. Sections 2 and 3 present the method in an abstract framework. Section 4 and 5 present two classes of stabilizations for the inf-sup condition in mixed problems. We present many examples, most arising from the discretization of flow problems. Section 6 presents examples in which the stabilizing terms is introduced to cure coercivity problems.

MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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