A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity

Abstract

We consider boundary value problems for the Laplace equation in three-dimensional multilayer domains composed of an infinite strip layer of finite height and a half-space containing a bounded cavity. The unknown (harmonic) function satisfies the Neumann boundary condition on the exterior boundary of the strip layer (i.e. at the bottom of the first layer), the Dirichlet, Neumann or Robin boundary condition on the boundary surface of the cavity and the corresponding transmission (matching) conditions on the interface layer boundary. We reduce this boundary value problem to a boundary integral equation over the boundary surface of the cavity by constructing Green's matrix for the corresponding transmission problem in the domain consisting of the infinite layer and the half-space (not with the cavity). This direct integral equation approach leads, for any of the above boundary conditions, to boundary integral equations with a weak singularity on the cavity. The numerical solution of this equation is realized by Wienert's [Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im R ³, Ph.D. thesis, University of Göttingen, Germany, 1990] method. The reduction of the problem, originally set in an unbounded three-dimensional region, to a boundary integral equation over the boundary of a bounded domain, is computationally advantageous. Numerical results are included for various boundary conditions on the boundary of the cavity, and compared against a recent indirect approach [R. Chapko, B.T. Johansson, and O. Protsyuk, On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R ³ with cavities, J. Numer. Appl. Math. (Kyiv) 105 (2011), pp. 4–18], and the results obtained show the efficiency and accuracy of the proposed method. In particular, exponential convergence is obtained for smooth cavities.

Keywords:

• boundary integral equations
• Galerkin method
• Gauss–Legandre quadratures
• Green's matrix
• semi-infinite layered domain
• Sinc-quadratures

2010 AMS Subject Classifications :

• 65R20
• 45B05
• 34B27

Acknowledgements

The authors would like to thank Dr Olha Ivanyshyn (Technische Universität Darmstadt) for helpful consultations related to the numerical implementation of the proposed method. The authors are also grateful for comments and suggestions made by the (anonymous) referees.