Circumventing the Babuscka-Brezzi condition in mixed finite element approximations of elliptic variational inequalities

Abstract In order to circumvent the Babuska-Brezzi (BB) condition in the finite element method with Lagrange multipliers on the boundary, for both elliptic variational equalities and inequalities, least-squares-like terms are added to the classical Galerkin method. The additional terms involve integrals over element interiors and mesh-parameter dependent coefficients. The resulting formulation retains consistency and attains convergence for arbitrary polynomial interpolations which are continuous for the primal variable and which may be continuous or discontinuous for the Lagrange multiplier.

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