**News**

–** Location**

La Nave Building 14, Via Edoardo Bonardi, 9 (access from via Ampère or from via Ponzio 31), 20133 Milano MI.

Sala del Consiglio, 7th floor. See here.

A link will be provided soon to all registered participants to attend the lectures online.

**– Covid regulations**

The event will be held according to the latest regulations regarding the Covid situation.

Will be admitted in presence only participants showing one of the following certifications:

1. green pass,

2. certificate of vaccination with Pfizer/Biontech, Moderna, Astrazeneca, Johnson (other vaccines do not comply with the Italian Covid regulation),

3. negative test not older than two days (can be performed also in pharmacies).

All participants must wear appropriate face cover (e.g., FFP2 or surgical mask).

– **Program**

(the abstracts can be found at the end of this page)

**Wednesday 24.11**

8:45-9:00 – Welcome

9:00-10:00 – Classical domain decomposition methods (M.J. Gander)

10:00-10:15 – Coffee break

10:15-11:15 – Introduction to Krylov methods (T. Vanzan)

11:15-12:15 – Dual-primal FETI domain decomposition methods (A. Klawonn)

12:15-14:15 – Lunch break

14:15-15:15 – Exercise session 1 (T. Vanzan)

**Thursday 25.11**

9:00-10:00 – Scalability and coarse space corrections (M.J. Gander)

10:00-10:15 – Coffee break

10:15-11:15 – An introduction to spectral coarse spaces (V. Dolean)

11:15-12:15 – Multilevel Spectral Domain Decomposition Methods (P. Bastian)

12:15-14:15 – Lunch break

14:15-15:15 – Exercise session 2 (A. Heinlein)

**Friday 26.11**

9:00-10:00 – Optimized Schwarz, algorithms and methods (L. Halpern)

10:00-10:15 – Coffee break

10:15-11:15 – Time parallelization methods and parareal algorithm (J. Salomon)

11:15-12:15 – About the use of Fourier arguments to study the convergence of Schwarz Waveform Relaxation Algorithms (V. Martin)

12:15-13:30 – Lunch break

13:30-14:30 – Domain decomposition methods for heterogeneous problems (T. Vanzan)

14:30-15:30 – Heterogeneous Domain Decomposition for Cardiovascular Problems (C. Vergara)

**General description**

We are pleased to announce the **Summer School on Advanced Domain Decomposition Methods** in Milan, Italy. The event is funded by the **European Mathematical Society** and will be held at the **Politecnico di Milano** **from 24.11.2021 to 26.11.2021**.

The goal of this summer school is to introduce basic and advanced domain decomposition methods to researchers interested in understanding these methods, and to give them detailed insight on how these methods reduce various error components and are used in practice.

The course will cover domain decomposition methods for various problems ranging from classical stationary partial differential equations (PDEs) to time-dependent problems, passing through PDEs with high-contrast coefficients and heterogeneous problems.

The school is suitable for young mathematicians (PhD students and highly motivated Master Students), who wish to either get in touch with domain decomposition methods, as well as young researchers and Post-docs who desire to strengthen and broaden their knowledge on advanced domain decomposition techniques.

**List of speakers**

**Bastian, Peter**(University of Heidelberg, Germany)**Dolean, Victorita**(University of Strathclyde, UK and University Côte d’Azur, France)**Gander, Martin J.**(University of Geneva, Switzerland)**Halpern, Laurence**(University of Paris 13, France)**Heinlein, Alexander**(TU Delft, Netherlands)**Klawonn, Axel**(University of Cologne, Germany)**Martin, Veronique**(Université de Picardie Jules Verne, France)**Salomon, Julien**(INRIA Paris, France)**Vanzan, Tommaso**(EPFL, Switzerland)**Vergara, Christian**(Politecnico di Milano, Italy)

**Registration** **(registration will close on 12.11.2021**)

The registration is **free of fees**. Participants can attend the lectures **either in presence or online** (further info will be available soon). To register send an email to Gabriele Ciaramella including the registration form (that you can download below) filled in and signed.

**Organizer**

**Ciaramella, Gabriele**(Politecnico di Milano, Italy)

**Contacts**

- gabriele.ciaramella_at_polimi.it (Gabriele Ciaramella)
- eventi-dmat_at_polimi.it (Events Office)

**Abstracts**

**Wednesday 24.11**

**Classical domain decomposition methods** (M. J. Gander)

Classical Schwarz methods, the Dirichlet-Neumann method, the Neumann-Neumann method, and Optimized Schwarz Methods. For each case, I will present the historical invention of the method and its basic functioning, including a convergence analysis for the Laplace problem, and show numerical examples.

**Introduction to Krylov methods **(T. Vanzan)

Krylov methods are classical iterative methods to solve large and sparse linear systems derived from the discretization of partial differential equations. The goal of this lecture is to provide a brief overview. As a reference method, we will mainly discuss the conjugate gradient method and introduce it as an improvement over the steepest descent method. We then discuss GMRES and provide a (very) brief overview of other famous Krylov methods. Next, we show that a Krylov method can be used to accelerate the convergence of a stationary iterative method, or equivalently that any stationary iterative method can be used as a preconditioner for a Krylov method. Finally, we discuss some tips on how to use preconditioning in practice.

**Dual-primal FETI domain decomposition methods** (A. Klawonn)

T.B.A.

**Exercise session 1** (T. Vanzan)

In the first practical session, the participants will implement some of the classical domain decomposition methods, namely the Schwarz method, the optimized Schwarz method, the Dirichlet-Neumann method and the Neumann-Neumann method, for the solution of a model problem. The goal is to deepen the understanding of how these methods work, analyze their dependence on some parameters (e.g. overlap or Robin/relaxationparameters), verify the convergence results seen in the first lecture, and experience how a Krylov method can accelerate convergence of a linear fixedpoint iteration. The participants will be provided with draft Matlab/Octave codes to complete during the session.

**Thursday 25.11**

**Scalability and coarse space corrections** (M. J. Gander)

I will start by showing when classical one level methods from the first lecture are

scalable without coarse correction, and when a coarse correction is needed. I will then show how to design a good coarse space correction for the Laplace model problem, and give numerical illustrations.

**An introduction to spectral coarse spaces** (V. Dolean)

In the presence of many subdomains, the performance of Schwarz algorithms, i.e., the iteration count will grow linearly with the number of subdomains in one direction. From a parallel computing point of view this translates into a lack of scalability. The latter can be achieved by adding a second level or a coarse space. This is closely related to multigrid methods and deflation methods from numerical linear algebra. The simplest coarse space, that of Nicolaides, is in fact a particular case of a more general class of spectral coarse spaces which are generated by vectors resulting from solving some local generalized eigenvalue problems. We present a theory of these two-level algorithms in a general variational setting as well as elements from the abstract theory of the two-level additive Schwarz methods (e.g., the concept of stable decomposition). These results are valid for symmetric positive definite PDEs although this was extended recently to midly indefinite problems (convection-diffusion and Helmholtz equation in a low frequency regime).

**Multilevel Spectral Domain Decomposition Methods** (P. Bastian)

Overlapping domain decomposition methods with spectral coarse spaces are introduced in the framework of subspace correction methods. On that basis, convergence results independent of the mesh size and coefficient variation are

obtained for discontinuous Galerkin discretizations of the elliptic model problem with full tensor as well as for extensions to more than two levels. Numerical results for highly heterogeneous problems with up to 16384 subdomains are shown.

**Exercise session 2** (A. Heinlein)

Schwarz methods are an algorithmic framework for a large class of domain decomposition methods. The software FROSch (Fast and Robust Overlapping Schwarz), which is part of the Trilinos package ShyLU, provides a highly scalable implementation of the Schwarz framework, and the resulting solvers are based on the construction and combination of the relevant Schwarz operators. FROSch currently focusses on Schwarz operators that are algebraic in the sense that they can be constructed from a fully assembled, parallel distributed matrix. This is facilitated by the use of extension-based coarse spaces, such as generalized Dryja-Smith-Widlund (GDSW) type coarse spaces.

In this lab session, the FROSch software framework will be introduced, and its usage will be explained based on simple model problems. The examples provided will allow investigating the influence of important algorithmic aspects of Schwarz methods, such as the variation of the width of the overlap or adding a coarse level, on the convergence of a preconditioned Krylov solver.

**Friday 26.11**

**Optimized Schwarz, algorithms and methods** (L. Halpern)

The Schwarz algorithm (Schwarz, 1872) is an overlapping domain decomposition algorithm applied to solving partial differential equations, whose properties are now well-known and widely used in the context of parallel computations. But the convergence of low frequency components deteriorates with the size of the overlap.

Optimized Schwarz algorithms have been developed in the last twenty years (starting with Caroline Japhet’s thesis, 1998), and improve drastically the convergence of the method. They modify the transmission conditions at the interfaces, using Robin or Ventcel operators. Through Fourier-type techniques, the best Robin coefficient is obtained through of a (possibly) complex best approximation problem. We present here a summary of advantages, techniques and results in various situations, including time dependent problems, optimal control problems, and coarsening.

**About the use of Fourier arguments to study theconvergence of Schwarz Waveform Relaxation Algorithms** (V.Martin)

Schwarz Methods and Schwarz Waveform Relaxation methods (SWR) give powerful algorithms to approximate the solutions of PDEs on large domains. The first one applies to stationary PDEs while the second one treats time dependant PDEs. To study the convergence speed of such algorithms, the Fourier transform is usually used: the convergence factor is given in Fourier variables. This strategy has been first developed for steady problems and rapidly this tool has also been used for unsteady equations.

In this talk, we investigate the question of the relevance to use Fourier arguments to study the SWR algorithms. In the case of the SWR algorithm with Dirichlet transmission conditions, we study the effect of the algorithm when agiven frequency

is introduced.

**Time parallelization methods and parareal algorithm** (J. Salomon)

After a review of time parallelization methods to approximate solutions of dynamical systems, we will focus on the parareal algorithm. We will recall the properties of this algorithm and detail its convergence analysis. The performance of this method will be illustrated by some numerical results. Finally, we will present some extensions of the parareal algorithm to optimal control and data assimilation problems.

**Domain decomposition methods for heterogeneous problems** (T.Vanzan)

Domain decomposition methods are well-known algorithms to solve partial differential equations. The solution strategy consists in decompose theproblem into hundreds, thousands, or even millions of smaller subproblems,that can be solved efficiently in parallel.In this lecture, we consider domain decomposition methods from a differentperspective. We focus on multiphysics problems, that is problems whose mathematical description requires the coupling of two or more different equations. Examples are fluid-structure interactions, filtration of fluids, heat conduction, oceanic-atmosphere coupling, contaminant transport and many others. In such problems, one usually deals with a small number of subdomains, as the decomposition of the computational domain is naturally provided by the physics.

We will discuss how nonoverlapping domain decomposition methods are useful decoupled approaches to solve such multiphysics problems, and how to accelerate their convergence through a proper choice of the coupling conditions.

**Heterogeneous Domain Decomposition for cardiovascular problems** (C. Vergara)

In this lesson, we focus on numerical methods based on Domain Decomposition (DD) for heterogeneous coupling problems arising in the field of the cardiovascular system. After a brief review of the main issues related to Heterogeneous coupled problems, such as the interface continuity conditions and partitioned schemes for their solution, we describe some examples of cardiovascular problems. First, we detail the Fluid-Structure Interaction (FSI) problem arising between blood flow and artery vessel dynamics. Secondly, we address the cardiac perfusion problem where incompressible fluid-dynamics in the large coronaries is coupled with a Darcy problem in the myocardium. Finally, we introduce the electro-mechanical coupling for the descriprion of the heart function. Suitable algorithms for the numerical solution and some results in realistic 3D scenario are also provided for all the cases.