Interface Control Domain Decomposition (ICDD) Methods for Coupled Diffusion and Advection-Diffusion Problems

18/2013

Interface Control Domain Decomposition (ICDD) Methods for Coupled Diffusion and Advection-Diffusion Problems

Monday 29th April 2013

Discacciati, M.; Gervasio, P.; Quarteroni, A.

This paper is concerned with ICDD (Interface Control Domain Decomposition) method, a strategy introduced for the solution of partial differential equations (PDEs) in computational domains partitioned into subdomains that overlap. After reformulating the original boundary value problem with the introduction of new additional control variables, the unknown traces of the solution at internal subdomain interfaces, the determination of the latter is made possible by the requirement that the (a-priori) independent solutions in each subdomain undergo a minimization of a suitable cost functional. We illustrate the method on two kinds of boundary value problems, one homogeneous (an elliptic PDE), the other heterogeneous (a coupling between a second order advection-diffusion equation and a first order advection equation). We derive the associated optimality system, analyze its well posedness, and illustrate efficient algorithms based on the solution of the Schur-complement system restricted solely to the interface control variables. Finally, we validate numerically our method through a family of numerical tests and investigate the excellent convergence properties of our iterative solution algorithm.