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

Anton Evgrafov; Misha Marie Gregersen; Mads Peter Sørensen

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 45, Issue: 6, page 1059-1080
- ISSN: 0764-583X

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topEvgrafov, Anton, Gregersen, Misha Marie, and Sørensen, Mads Peter. "Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 45.6 (2011): 1059-1080. <http://eudml.org/doc/276352>.

@article{Evgrafov2011,

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.
},

author = {Evgrafov, Anton, Gregersen, Misha Marie, Sørensen, Mads Peter},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Topology optimization; finite volume methods; topology optimization; nonlinear elliptic equation; convergence; Laplace equation; steady state heat conduction; numerical example},

language = {eng},

month = {6},

number = {6},

pages = {1059-1080},

publisher = {EDP Sciences},

title = {Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients},

url = {http://eudml.org/doc/276352},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Evgrafov, Anton

AU - Gregersen, Misha Marie

AU - Sørensen, Mads Peter

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2011/6//

PB - EDP Sciences

VL - 45

IS - 6

SP - 1059

EP - 1080

AB -
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.

LA - eng

KW - Topology optimization; finite volume methods; topology optimization; nonlinear elliptic equation; convergence; Laplace equation; steady state heat conduction; numerical example

UR - http://eudml.org/doc/276352

ER -

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