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

• Published in: ESAIM. Mathematical modelling and numerical analysis Vol. 45; no. 6; pp. 1059 - 1080
• Main Authors: Evgrafov, Anton, Gregersen, Misha Marie, Sørensen, Mads Peter
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.11.2011
• Subjects: 49M05; 49M25; 65N08; 65N12; Algorithms; Boundary conditions; Calculus of variations and optimal control; Combinatorics; Combinatorics. Ordered structures; Convergence; Design optimization; Designs and configurations; Exact sciences and technology; Finite volume method; finite volume methods; Integrals; Laplace transforms; Mathematical analysis; Mathematical models; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Numerical methods in optimization and calculus of variations; Partial differential equations; Product development; Sciences and techniques of general use; Studies; Topology optimization; Grid pattern; Limit point; Optimization method; Laplace equation; Finite volume method; Algorithm; Steady state; Discretization method; Convergence; Finite element method; Lebesgue measure; Numerical analysis; Optimal solution; Variational calculus; Mathematical model; Mathematical programming
• ISSN: 0764-583X; 1290-3841
• DOI: 10.1051/m2an/2011012

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.