### Abstract

Regions containing internal boundaries such as composite materials arise in many applications.We consider a situation of a layered domain in IR^{3} containing a nite number of bounded cavities. The model is stationary heat transfer given by the Laplace equation with piecewise constant conductivity. The heat ux (a Neumann condition) is imposed on the bottom of the layered region and various boundary conditions are imposed on the cavities. The usual transmission (interface) conditions are satised at the interface layer, that is continuity of the solution and its normal derivative. To eciently calculate the stationary temperature eld in the semi-innite region, we employ a Green's matrix technique and reduce the problem to boundary integral equations (weakly singular) over the bounded surfaces of the cavities. For the numerical solution of these integral equations, we use Wienert's approach [20]. Assuming that each cavity is homeomorphic with the unit sphere, a fully discrete projection method with super-algebraic convergence order is proposed. A proof of an error estimate for the approximation is given as well. Numerical examples are presented that further highlights the eciency and accuracy of the proposed method.

Original language | English |
---|---|

Pages (from-to) | 4-18 |

Number of pages | 15 |

Journal | Journal of Numerical and Applied Mathematics |

Volume | 105 |

Issue number | 2 |

Publication status | Published - 2011 |

### Fingerprint

### Keywords

- semi-innite multilayer domain
- Green's matrix
- boundary integral equations
- weak singularities
- Galerkin method
- spherical functions
- Gauss-Legandre quadratures
- sinc-quadratures

### Cite this

*Journal of Numerical and Applied Mathematics*,

*105*(2), 4-18.

}

*Journal of Numerical and Applied Mathematics*, vol. 105, no. 2, pp. 4-18.

**On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R3 with cavities.** / Chapko, R.S.; Johansson, B.T.; Protsyuk, O.B.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R3 with cavities

AU - Chapko, R.S.

AU - Johansson, B.T.

AU - Protsyuk, O.B.

PY - 2011

Y1 - 2011

N2 - Regions containing internal boundaries such as composite materials arise in many applications.We consider a situation of a layered domain in IR3 containing a nite number of bounded cavities. The model is stationary heat transfer given by the Laplace equation with piecewise constant conductivity. The heat ux (a Neumann condition) is imposed on the bottom of the layered region and various boundary conditions are imposed on the cavities. The usual transmission (interface) conditions are satised at the interface layer, that is continuity of the solution and its normal derivative. To eciently calculate the stationary temperature eld in the semi-innite region, we employ a Green's matrix technique and reduce the problem to boundary integral equations (weakly singular) over the bounded surfaces of the cavities. For the numerical solution of these integral equations, we use Wienert's approach [20]. Assuming that each cavity is homeomorphic with the unit sphere, a fully discrete projection method with super-algebraic convergence order is proposed. A proof of an error estimate for the approximation is given as well. Numerical examples are presented that further highlights the eciency and accuracy of the proposed method.

AB - Regions containing internal boundaries such as composite materials arise in many applications.We consider a situation of a layered domain in IR3 containing a nite number of bounded cavities. The model is stationary heat transfer given by the Laplace equation with piecewise constant conductivity. The heat ux (a Neumann condition) is imposed on the bottom of the layered region and various boundary conditions are imposed on the cavities. The usual transmission (interface) conditions are satised at the interface layer, that is continuity of the solution and its normal derivative. To eciently calculate the stationary temperature eld in the semi-innite region, we employ a Green's matrix technique and reduce the problem to boundary integral equations (weakly singular) over the bounded surfaces of the cavities. For the numerical solution of these integral equations, we use Wienert's approach [20]. Assuming that each cavity is homeomorphic with the unit sphere, a fully discrete projection method with super-algebraic convergence order is proposed. A proof of an error estimate for the approximation is given as well. Numerical examples are presented that further highlights the eciency and accuracy of the proposed method.

KW - semi-innite multilayer domain

KW - Green's matrix

KW - boundary integral equations

KW - weak singularities

KW - Galerkin method

KW - spherical functions

KW - Gauss-Legandre quadratures

KW - sinc-quadratures

UR - http://jnam.lnu.edu.ua/pdf/y2011_no2(105)_art02_chapko_johansson_protsyuk.pdf

M3 - Article

VL - 105

SP - 4

EP - 18

JO - Journal of Numerical and Applied Mathematics

JF - Journal of Numerical and Applied Mathematics

SN - 0868-6912

IS - 2

ER -