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A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. (English) Zbl 0923.65064

Authors’ abstract: For the groundwater flow problem (which corresponds to the Darcy flow model), we show how to produce a scheme with one unknown per element, starting from a mixed formulation discretized with the Raviart Thomas triangular elements of lowest order. The aim is here to obtain a new formulation with one unknown per element by elimination of the velocity variables \({\mathbf q} = -k\text{ grad }P\), without any restriction concerning the computation of the velocity field. In the first part, we describe the triangular mixed finite element method used for solving Darcy’s and mass balance equations.
In the second part, we study the elliptic-parabolic problem; we describe the new formulation of the problem in order to use mixed finite elements (MFE) with less unknowns without any specific numerical integration. Finally, we compare the computational effort of the MFE method with the new formulation for different triangulations using numerical experiments. In this work, we show that the new formulation can be seen as a general formulation which can be equivalent to the finite volume or the finite difference methods in some particular cases.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
35K15 Initial value problems for second-order parabolic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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