Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with a variable coefficient

Abstract

In this paper, a numerical implementation of a direct united boundary-domain integral equation (BDIE) related to the Neumann boundary value problem for a scalar elliptic partial differential equation with a variable coefficient is discussed. The BDIE is reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretization of the BDIEs with quadrilateral domain elements leads to a system of linear algebraic equations (discretized BDIE). Then, the system is solved by LU decomposition and Neumann iterations. Convergence of the iterative method is discussed in relation to the distribution of eigenvalues of the corresponding discrete operators calculated numerically.

Keywords:

• boundary-domain integral equations
• numerical solution
• iterative methods
• spectrum
• eigenvalues

2010 AMS Subject Classifications :

• 65N38
• 31A10
• 45A05
• 45C05
• 65R20

Acknowledgements

The work was supported by the grant EP/H020497/1 ‘Mathematical analysis of localized boundary-domain integral equations for BVPs with variable coefficients’ of the EPSRC, UK.