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Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients

Published online by Cambridge University Press:  28 June 2011

Anton Evgrafov ,
Misha Marie Gregersen  and
Mads Peter Sørensen
Anton Evgrafov
Affiliation:
Department of Mathematics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark. a.evgrafov@mat.dtu.dk
Misha Marie Gregersen
Affiliation:
Department of Mathematics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark. a.evgrafov@mat.dtu.dk
Mads Peter Sørensen
Affiliation:
Department of Mathematics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark. a.evgrafov@mat.dtu.dk
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Abstract

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.

Keywords

Topology optimizationfinite volume methods
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 6 , November 2011 , pp. 1059 - 1080
Copyright
© EDP Sciences, SMAI, 2011

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