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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:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J25Second order elliptic equations, boundary value problems
65N12Stability and convergence of numerical methods (BVP of PDE)
65N15Error bounds (BVP of PDE)