## Expanded mixed finite element methods for linear second-order elliptic problems. I.(English)Zbl 0910.65079

This is the first paper of a series in which the author presents a new mixed formulation for the numerical solution of second-order elliptic problems. This new formulation expands the standard mixed formulation in the sense that three variables are explicitly treated: the scalar unknown, its gradient, and its flux (the coefficient times the gradient). Based on this formulation, mixed finite element approximations of the second-order elliptic problems are considered. Optimal order error estimates in the $$L^p$$- and $$H^{-s}$$-norms are obtained for the mixed approximations. Various implementation techniques for solving systems of algebraic equations are discussed. A postprocessing method for improving the scalar variable is analyzed, and superconvergent estimates in the $$L^p$$-norm are derived. The mixed formulation is suitable for the case where the coefficient of differential equations is a small tensor and does not need to be inverted.
Reviewer: K.Najzar (Praha)

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs

Zbl 0910.65080
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### References:

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