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Partial Differential Equations - Modelling and ... - ResearchGate

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Mixed Finite Element Methods on Polyhedral<br />

Meshes for Diffusion <strong>Equations</strong><br />

Yuri A. Kuznetsov<br />

Department of Mathematics, University of Houston, 651 Philip G. Hoffman Hall,<br />

Houston, TX 77204–3008, USA kuz@math.uh.edu<br />

Summary. In this paper, a new mixed finite element method for the diffusion<br />

equation on polyhedral meshes is proposed. The method is applied to the diffusion<br />

equation on meshes with mixed cells when all the coefficients <strong>and</strong> the source function<br />

may have discontinuities inside polyhedral mesh cells. The resulting discrete equations<br />

operate only with the degrees of freedom for normal fluxes on the boundaries<br />

of cells <strong>and</strong> one degree of freedom per cell for the solution function.<br />

Key words: Diffusion equation, mixed finite element method, polyhedral<br />

meshes, mixed cells<br />

1 Introduction<br />

In this paper, we propose a new mixed finite element method for the diffusion<br />

equation on general polyhedral meshes in the case when the coefficients of the<br />

equation <strong>and</strong> the source function may have strong discontinuities inside mesh<br />

cells. Such mesh cells are called mixed ones. The major idea of the method is<br />

reported in [Kuz05]. This work is a natural extension of the method in [Kuz06]<br />

to 3D diffusion equations.<br />

The discretization method consists of several steps. At the first step, we<br />

partition each polyhedral cell into polyhedral subcells assuming that inside<br />

each subcell the coefficients <strong>and</strong> the source function are relatively smooth.<br />

Then, in each subcell we impose a local conforming tetrahedral mesh subject<br />

to a structure of the neighboring subcells. The subcell tetrahedral meshes are<br />

not required to be conforming on the interfaces between subcells. A special<br />

finite element subspace of H div (Ω) is invented, <strong>and</strong> the classical mixed finite<br />

element method [BF91, RT91] is used for discretization of the diffusion equation<br />

with the Neumann boundary condition. At the final step, the interior<br />

(with respect to the boundaries of polyhedral mesh cells) degrees of freedom<br />

for the normal fluxes <strong>and</strong> for the solution function are eliminated, <strong>and</strong> a new

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