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**Equivalence between lowest-order mixed finite element and multi-point finite volume methods. Derivation, properties and numerical experiments.**
*(English)*
Zbl 1180.65154

Handlovičová, Angela (ed.) et al., ALGORITMY 2005. 17th conference on scientific computing, Vysoké Tatry – Podbanské, Slovakia, March 13–18, 2005. Proceedings of contributed papers and posters. Bratislava: University of Technology, Faculty of Civil Engineering, Department of Mathematics and descriptive Geometry (ISBN 80-227-2192-1). 103-112 (2005).

Summary: We consider the lowest-order Raviart-Thomas mixed finite element method tor elliptic diffusion problems on simplicial meshes in two or three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily eliminate the flux unknowns, which implies an equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive definite for a large class of problems, but in general nonsymmetric.

We next show that these ideas also apply to mixed and upwind-mixed finite element discretizations of nonlinear parabolic convection-reaction-diffusion problems. We finally present a set of numerical experiments confirming important computational savings while using the equivalent finite volume form of the lowest-order mixed finite element method.

For the entire collection see [Zbl 1175.65002].

We next show that these ideas also apply to mixed and upwind-mixed finite element discretizations of nonlinear parabolic convection-reaction-diffusion problems. We finally present a set of numerical experiments confirming important computational savings while using the equivalent finite volume form of the lowest-order mixed finite element method.

For the entire collection see [Zbl 1175.65002].

### MSC:

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

65N08 | Finite volume methods for boundary value problems involving PDEs |

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

35K57 | Reaction-diffusion equations |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

### Keywords:

multi-point finite volume scheme; numerical experiments; lowest-order Raviart-Thomas mixed finite element method; saddle-point problem; multi-point finite volume method; elliptic diffusion equation; nonlinear parabolic convection-reaction-diffusion equation
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\textit{M. Vohralík}, in: ALGORITMY 2005. 17th conference on scientific computing, Vysoké Tatry, Podbanské, Slovakia, March 13--18, 2005. Proceedings of contributed papers and posters. Bratislava: University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry. 103--112 (2005; Zbl 1180.65154)

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