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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 elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 35J25 Second order elliptic equations, boundary value problems 65N12 Stability and convergence of numerical methods (BVP of PDE) 65N15 Error bounds (BVP of PDE)
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