Optimized Schwarz methods for the coupling of cylindrical geometries along the axial direction

¹ Dipartimento di Ingegneria gestionale, dell’informazione e della produzione, Università degli Studi di Bergamo, Viale Marconi 5, 24044 Dalmine (BG), Italy
² MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy

^* Corresponding author: christian.vergara@polimi.it

Received: 28 February 2017
Accepted: 25 May 2018

Abstract

In this work, we focus on the Optimized Schwarz Method for circular flat interfaces and geometric heterogeneous coupling arising when cylindrical geometries are coupled along the axial direction. In the first case, we provide a convergence analysis for the diffusion-reaction problem and jumping coefficients and we apply the general optimization procedure developed in Gigante and Vergara (Numer. Math. 131 (2015) 369–404). In the numerical simulations, we discuss how to choose the range of frequencies in the optimization and the influence of the Finite Element and projection errors on the convergence. In the second case, we consider the coupling between a three-dimensional and a one-dimensional diffusion-reaction problem and we develop a new optimization procedure. The numerical results highlight the suitability of the theoretical findings.