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**A characterization of hybridized mixed methods for second order elliptic problems.**
*(English)*
Zbl 1084.65113

The authors give a new characterization of the approximate solutions by hybridized mixed methods for second order self-adjoint elliptic problems. Here hybridization means the introduction of a Lagrange multiplier on the edges to make the original continuous unknown functions be discontinuous across the interelement edges. This characterization is used to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization.

In particular, the matrix equation becomes to be similar to those of the primary finite element method using the displacement variables only, so that it can be dealt with various standard solvers for linear algebraic equations. Furthermore, necessary and sufficient conditions are obtained, under which the multipliers of the Raviart-Thomas and the Brezzi-Douglas-Marini methods of similar order are identical.

The present analysis is effective for both practical implementation and theoretical understanding of some mixed finite element methods.

In particular, the matrix equation becomes to be similar to those of the primary finite element method using the displacement variables only, so that it can be dealt with various standard solvers for linear algebraic equations. Furthermore, necessary and sufficient conditions are obtained, under which the multipliers of the Raviart-Thomas and the Brezzi-Douglas-Marini methods of similar order are identical.

The present analysis is effective for both practical implementation and theoretical understanding of some mixed finite element methods.

Reviewer: Fumio Kikuchi (Tokyo)

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |