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{\bf Finite Volume Methods for Steady-State and Transient Problems} \\
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R.D.~Lazarov \\
 Dept. of Mathematics, Texas A\&M University, College Station, Texas
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We consider elliptic and parabolic differential equations subject to
Dirichlet and/or Neumann boundary conditions and derive locally
conservative schemes based on the approximation of the balance
equation for a set of control volumes. First we introduce the triangulation
$T_h$ by splitting the domain into triangles and discuss two basic
approximation strategies.

In the first approach, called finite volume element method, we introduce
the unknown values of the solution at the vertices of
the triangulation $T_h$ as degrees of freedom and
form the space $V_h$  of continuous piece-wise
linear functions over the triangulation $T_h$.
Next, we introduce the dual triangulation $T_h^*$
of finite volumes around each vertex in $T_h$. The finite volume
element scheme is a Petrov-Galerkin method with solution space $V_h$ and
test space $V_h^*$ of piece-wise constants over the finite volumes in $T_h^*$.

In the  the second approach the degrees of freedom are the values of the
unknown solution at the ortho-centers of the triangles from $T_h$ and the
dual grid consists of the Voronoi volumes (formed by the
the perpendicular bisectors of the edges in $T_h$). The balance
equation is written for $T_h$ and leads to discretization which involves the
values at the ortho-centers. In this case we have to restrict the partition
$T_h$ to Delauney triangles. For problems without mixed derivatives this
approach leads to finite difference approximations with harmonic averaging
of the coefficients.

We study the stability, the approximation, and the convergence
of these schemes  in the natural discrete energy norm.


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