Second kind boundary integral equation for multi-subdomain diffusion problems

Xavier Claeys (UPMC)

We study elliptic boundary value problems where the coefficients are piecewise constant with respect to a partition of space into Lipschitz subdomains, focusing on the case of jumping coefficients arising in the principal part of the partial differential operator. We propose a boundary integral equation of the second kind posed on the interfaces of the partition, and involving only one unknown trace function at each interface. We provide a detailed analysis of the corresponding integral operator, proving well-posedness. We also present numerical results that exhibit a systematically stable condition number for the associated Galerkin matrices, so that GMRES seems to enjoy fast convergence independent of the mesh resolution.