We present a convergence analysis of a cell-based finite volume (FV)
discretization scheme applied to a problem of control in the
coefficients of a generalized Laplace equation modelling, for
example, a steady state heat conduction.
Such problems arise in applications dealing with geometric optimal
design, in particular shape and topology optimization, and are most
often solved numerically utilizing a finite element approach.
Within the FV framework for control in the coefficients problems
the main difficulty we face is the need to analyze the convergence
of fluxes defined on the faces of cells, whereas the
convergence of the coefficients happens only with respect to the
“volumetric” Lebesgue measure.
Additionally,
depending on whether the stationarity conditions are stated for the
discretized or the original continuous problem, two distinct
concepts of stationarity at a discrete level arise.
We provide characterizations of limit points, with respect to FV
mesh size, of globally optimal solutions and two types of
stationary points to the discretized problems.
We illustrate the practical behaviour of our cell-based FV
discretization algorithm on a numerical example.