Partial Differential Equations - Modelling and ... - ResearchGate

28 Yu.A. Kuznetsov
degree of freedom per mesh cell for the solution function is defined. The final
system of discrete equations has the same structure as for the classical mixed
FE method.
The paper is organized as follows. In Section 2, we formulate the problem
and requirements for the discretization. In Section 3, we describe partitionings
of mesh cells into subcells and polyhedral meshes to be used for the
discretization. We also propose a special finite element subspace of H div (Ω)
for the mixed finite element method. Finally, in Section 4, we describe a condensation
procedure for the underlying algebraic system and transform the
condensed system into the standard form which is typical for the classical
finite element method on simplicial meshes. In the final part of Section 3, we
propose an alternative discretization method. In Remark 2 of Section 4, we
prove that this discretization method is equivalent to the “div-const” mixed
finite element method invented and investigated in [KR03, KR05].
2 Problem Formulation
We consider the diffusion equation
with the Neumann boundary condition
− div(a grad p)+cp = f in Ω (1)
(a grad p) · n =0 on∂Ω (2)
where Ω is a polyhedral domain in R 3 with the boundary ∂Ω, a = a(x)
is a symmetric positive definite 3 × 3 matrix (diffusion tensor) for any x =
(x 1 ,x 2 ,x 3 ) ∈ Ω, c is a nonnegative function, f is a given source function, and
n is the outward unit normal to ∂Ω. The domain Ω is partitioned into m
open non-overlapping simply connected polyhedral subdomains Ω k with the
boundaries ∂Ω k , k = 1,m, i.e. Ω = ⋃ m
k=1 Ω k. For the sake of simplicity, we
assume that in each of the subdomains Ω k the matrix a has constant entries
and the coefficient c is a nonnegative constant, k = 1,m. We naturally assume
that in the case c ≡ 0inΩ the compatibility condition
∫
f dx = 0 (3)
Ω
holds.
In this paper, we consider problem (1), (2) in the form of the first order
system
a −1 u +gradp =0 inΩ,
− div u − cp = −f in Ω,
(4)
u · n =0 on∂Ω,
where u is said to be the flux vector function.