A non-overlapping **optimized** **Schwarz** **method** for the heat equation with non linear boundary conditions and with applications to de-icing
L. Bennani a,∗ , P. Trontin a , R. Chauvin b , P. Villedieu a
a ONERA / DMPE, Universit´ e de Toulouse, F-31055 Toulouse - France b CEA / DAM / DIF, F-91680 Bruy` eres-le-Chˆ atel - France

The study of is done for a decomposition of into two infinite domains. A Fourier transform is applied with respect to the tangential variables to the interface (artificial boundary sep- arating the two domains). The resulting local equations can be solved leading to the formulation of an iterative process applied to the interface variables. Then, we obtain the reduction factor of the error as a function of the Fourier variable and the pa- rameters involved in the interface conditions. In order to obtain the best possible convergence rate, one needs to optimize this quantity with respect to the parameters, for the range of possible spatial frequencies that can be represented on a given mesh. In the sequel, we treat the cases of zero order boundary conditions where we take in (2) equal to zero that is the case of gener- alized impedance conditions. Two possibilities are considered: and . It has been proved that are equal to where and are reported in Table I. When the mesh parameter is small, the maximum numerical frequency that can be represented on the mesh is estimated by where is a constant. We also define such that in order to exclude the frequency from the optimization process and this frequency being treated by the Krylov **method** (see also [6] for details).

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The **Optimized** **Schwarz** **Method** was originally introduced in [19, 20, 21, 22] considering general non-overlapping partition of the computational domain and constant scalar impedance coefficients. Although, in such a general geometrical setting, OSM with scalar impedance was proved to converge, no assessment was provided as regards the rate of convergence. In practice, the convergence could be slow. This was improved by Collino and Joly in [16, 31, 15] where the authors proposed operator valued self-adjoint positive impedance coefficients and could establish geometric convergence of the **method** assuming that the subdomain partition does not involve any cross point i.e. point of adjacency of three interfaces (or one interface meeting the boundary of the compuational domain), see Fig.1 above. In another series of contributions Antoine, Geuzaine and their collaborators [2, 25, 24, 5, 39] considered the case of impedance coefficients approaching appropriate Dirichlet-to-Neumann maps and obtained fastly converging numerical methods. Here also, the numerical methods were observed to be of good quality only when the subdomain partition does not contain any cross-point.

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effective transmission conditions, which can take the physics of the problem at hand into account, see [11, 12] and references therein. This property is especially important for anisotropic diffusion problems, which behave very dif- ferently at interfaces depending on the orientation of the diffusion. Similarly when discretizing anisotropic diffusion problems, the numerical scheme must be suitable for high anisotropy, and discrete duality finite volume (DDFV) meth- ods have this property, even in the case of discontinuous anisotropic diffusion, see [17, 4, 5] and references therein. We are therefore interested in **optimized** **Schwarz** **method** which are discretized using DDFV schemes. DDFV schemes belong to the class of discretization methods which preserve certain geometric properties of the underlying differential operators, like mimetic finite difference methods [18, 6], gradient methods [8], or discrete variational derivative methods [10], see also finite element exterior calculus [2]. DDFV methods are thus part of the effort to lead the field of geometric numerical integration, which reached a certain maturity for ordinary differential equations [16] to the area of partial differential equations.

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reduction of 10 −8 obtained with the classical and **optimized** **Schwarz** algorithm
the theoretical result in Theorem 3: the curves fit nicely the dependence on h predicted, i.e they behave like h −0.5 . We also see the tremendous improve-
ment of the **optimized** **Schwarz** **method** over the classical **Schwarz** **method**, which nevertheless performs a bit better than predicted in Corollary 1, the dependence on h measured is O(h −2 ), instead of O(h −3 ); for an explanation,

[19] M. J. Gander. **Optimized** **Schwarz** methods for Helmholtz problems. In Thirteenth international conference on domain decomposition, pages 245–252. CIMNE, Barcelona, Spain, 2001.
[20] M. J. Gander, L. Halpern, and F. Magoul`es. An **optimized** **Schwarz** **method** with two-sided Robin transmission conditions for the Helmholtz equation. Int. J. for Num. Meth. in Fluids, 55(2):163–175, 2007. [21] M.J. Gander, F. Magoul`es, and F. Nataf. **Optimized** **Schwarz** methods

This paper is organized as follows: in Section 2, we present Maxwell’s equations and a reformulation thereof with characteristic variables used in our analysis. In Section 3, we treat the case of time harmonic solutions. We show that the classical **Schwarz** **method** for Maxwell’s equations, which uses characteristic Dirichlet transmis- sion conditions between subdomains is convergent even without overlap. Exploiting a parallel with an **optimized** **Schwarz** **method** applied to an Helmholtz equation allows us to develop an entirely new hierarchy of **optimized** **Schwarz** methods for Maxwell’s equations with greatly enhanced performance, both with and without overlap. Similar equivalence has been presented in [13] for the Cauchy-Riemann equations. In Section 4, we present and analyze the corresponding hierarchy of **optimized** **Schwarz** methods for time discretizations of Maxwell’s equations. We then show in Section 5 numerical experiments in two and three spatial dimensions, both for the time harmonic and time discretized case, which illustrate the performance of the new **optimized** **Schwarz** methods for Maxwell’s equations. We also include as an application the cooking of a chicken in a microwave oven, a problem with variable coefficients. In Section 6, we summarize our findings and conclude with an outlook on future research directions.

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Theorem 5 and Theorem 7 contain the surprising result that in the presence of jumps in the coefficients, it is possible to obtain an **optimized** **Schwarz** **method** for Maxwell’s equations with convergence factor that does not deteriorate when the mesh parameter h goes to zero, even without overlap. In the first parts of each the- orem, we even see the convergence is independent of the jump in the coefficients. In the case of µ 1 = µ 2 in Theorem 6 however, the convergence factor depends on h and deteriorates as h goes to zero, as in the case in [7] when also ε 1 = ε 2 .

Espaces grossiers pour les méthodes de décomposition de domaine avec conditions d’interface optimisées
Abstract
The objective of this thesis is to design an efficient domain decomposition **method** to solve solid and ﬂuid mechanical problems, for this, **Optimized** **Schwarz** methods (OSM) are considered and revisited. The op- timized **Schwarz** methods were introduced by P.L. Lions. They consist in improving the classical **Schwarz** **method** by replacing the Dirichlet interface conditions by a Robin interface conditions and can be ap- plied to both overlapping and non overlapping subdomains. Robin conditions provide us an another way to optimize these methods for better convergence and more robustness when dealing with mechanical problem with almost incompressibility nature. In this thesis, a new theoretical framework is introduced which consists in providing an Additive **Schwarz** **method** type theory for **optimized** **Schwarz** methods, e.g. Lions’ algorithm. We define an adaptive coarse space for which the convergence rate is guaranteed regardless of the regularity of the coefficients of the problem. Then we give a formulation of a two-level preconditioner for the proposed **method**. A broad spectrum of applications will be covered, such as in- compressible linear elasticity, incompressible Stokes problems and unstationary Navier-Stokes problem. Numerical results on a large-scale parallel experiments with thousands of processes are provided. They clearly show the effectiveness and the robustness of the proposed approach.

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The purpose of this article is to define a general framework for building adap- tive coarse space for OSM methods for decomposition into overlapping subdo- mains. We prove that we can achieve the same robustness that what was done for **Schwarz** [35] and FETI-BDD [36] domain decomposition methods with so called GenEO (Generalized Eigenvalue in the Overlap) coarse spaces. Compared to these previous works, we have to introduce SORAS (symmetrized ORAS) a non standard symmetric variant of the ORAS **method** as well as two generalized eigenvalue problems. As numerical results will show in § 6.3, the **method** scales very well for saddle point problems such as highly heterogeneous nearly incom- pressible elasticity problems as well as the Stokes system. More precisely, in § 2, we give a short presentation of the current theory for the additive **Schwarz** **method**. Then, in section 3, we present algebraic variants to the P.L. Lions’ domain decomposition **method**. In § 4, we build a coarse space so that the two- level SORAS **method** achieves a targeted condition number. In § 5, the **method** is applied to saddle point problems.

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We are also interested in this article about the effectivness of the **method** on parallel computers. Another import issue for the **method** is therefore the scal- ability. As we know, without additional considerations, the more subdomains are used to decomposed (a 0 , b 0 ), the more iterations are required for SWR al- gorithm to reach convergence. Thus, the total computation time could hardly decrease significantly. In this paper, we propose two solutions: a new scalable algorithm if the potential is independent of time and a preconditioned algorithm for general potentials.

Conclusion
De par sa géométrie particulière et son rôle dans le problème de Nevanlinna-Pick spec- tral (NPS), le bidisque symétrisé a suscité l’intérêt de plusieurs mathématiciens au cours des dernières années. Cet ensemble jouit d’une structure simple caractérisée en termes d’un système de coordonnées intuitif. Une étude analytique et géométrique du bidisque symétrisé mène aux systèmes de **Schwarz**-Pick, des ensembles de métriques intimement liés aux fonctions holomorphes. Les pseudodistances de Carathéodory et de Kobayashi jouent un rôle de premier plan, celles-ci formant respectivement le plus petit et le plus grand système. Il est possible de trouver une formule explicite pour la pseudodistance de Carathéodory sur le bidisque symétrisé et ce, en exploitant la structure particulière de cet ensemble et en considérant des opérateurs bien choisis. La pseudodistance de Kobayashi est de plus égale à celle de Carathéodory sur ce domaine en particulier. Pour le démontrer, il faut adopter une stratégie complètement dif- férente. L’idée est de ramenener le calcul de la pseudodistance de Kobayashi à un problème d’interpolation sur le disque unité, c’est-à-dire au problème de Nevanlinna-Pick classique. Que les pseudodistances de Carathéodory et de Kobayashi soient égales sur le bidisque sy- métrisé est une découverte mathématique des moins banales. C’est pour cette raison que cet ensemble a été au coeur de plusieurs recherches au cours des dernières années. En plus d’appa- raître naturellement dans le NPS, la riche géométrie du bidisque symétrisé a su capter d’autant plus l’attention de plusieurs chercheurs dans le domaine.

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e.g. [8, 21, 28, 26, 6, 7]. However, when the discontinuities are along subdomain interfaces, classical results break down. This is what we work to improve.
In previous work, [23], two of the authors proposed the construction of a coarse subspace, which leads to a two-level **method** that is observed to be robust with respect to heterogeneous coefficients for fairly arbitrary domain decompositions, e.g. provided by an automatic graph partitioner such as METIS or SCOTCH [19, 5]. This **method** was extensively studied from a numerical point of view in [24]. The construction is based on the low-frequency modes asso- ciated with the Dirichlet-to-Neumann (DtN) map on each subdomain. After obtaining the eigenvectors associated with the near-kernel components of the DtN operator, we use their harmonic extensions to the whole subdomain to build the coarse grid. With this **method**, even for discontinuities along (rather than across) the subdomain interfaces, the iteration counts are robust to arbitrarily large jumps of the coefficients leading to a very efficient, au- tomatic **method** for these kinds of problems. It is also suitable for parallel implementation. Similar ideas to build stable coarse spaces, based on the solution of eigenvalue problems on the whole subdomain, have also been presented in other papers on **Schwarz** methods [15, 10], as well as in earlier work on algebraic multigrid methods [2, 4].

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Fig. 2.1: Illustration of the residual when RAS is used as a nonlinear solver (left), or as a preconditioner for Newton’s **method** (right).
iterations and using RASPEN as a preconditioner for Newton when solving the Forch- heimer equation with 8 subdomains from the numerical section. We observe that the residual of the non-linear RAS **method** is concentrated at the interfaces, since it must be zero inside the subdomains by construction. Thus, when Newton’s **method** is used to solve (2.6), it only needs to concentrate on reducing the residual on a small num- ber of interface variables. This explains the fast convergence of RASPEN shown on the right of Figure 2.1, despite the rather slow convergence of the underlying RAS iteration.

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1 Introduction
We present an open-source framework for testing **Schwarz**-type domain decomposition methods for time-harmonic wave problems. Such problems are known to be computationally challeng- ing, especially in the high-frequency regime. Among the various approaches that can be used to solve such problems, the Finite Element **Method** (FEM) with an Absorbing Boundary Con- dition (ABC) or a Perfectly Matched Layer (PML) is widely used for its ability to handle complex geometrical configurations and materials with non-homogeneous properties. However, the brute-force application of the FEM in the high-frequency regime leads to the solution of very large, complex-valued and possibly indefinite linear systems [45]. Direct sparse solvers do not scale well for such problems, and Krylov subspace iterative solvers exhibit slow convergence or diverge, while efficiently preconditioning proves difficult [28]. Domain decomposition meth- ods provide an alternative, iterating between subproblems of smaller sizes, amenable to sparse direct solvers [55].

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Domain decomposition **method** (DDM) is a general strategy for solving high-dimensional PDEs. Among DDMs, the **Schwarz** Waveform Relaxation (SWR) **method** is a popular algo- rithm for the numerical computation of evolution equations [13–19], in particular wave-like equations. SWR methods are characterized by the choice of the Transmission Conditions (TC) at the subdomain interfaces: Classical SWR is based on Dirichlet TC, Robin SWR uses Robin TC, Optimal SWR is related to transparent TC, and quasi-optimal SWR is based on accurate absorbing TC. **Optimized** SWR usually refers to Robin SWR, where the Robin parameters are **optimized** to ensure the fastest convergence possible of the algorithm. The latter then offers a good balance between fast convergence rate and efficient IBVP solver. In this paper, we are specifically interested in **Optimized** SWR methods. We now briefly describe the **Schwarz** Waveform Relaxation algorithms and set the problem of the selection of an **optimized** choice of the Robin parameter in the transmission conditions. Consider a d-dimensional evolution partial differential equation Pφ = f in the spatial domain Ω ⊆ R d , and time domain ( t

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position. In this chapter, we will explore the second approach to define **Schwarz** preconditioners
and thus, we will adapt the second point of view on **Schwarz** **method** to BEM matrices.
It is quite counter intuitive to apply techniques based on locality as Additive **Schwarz** **Method** (ASM), to non-local problems as Boundary Integral Equations (BIE). But several works followed this idea and an intuition of why it works anyway would be that because of the singularity of the kernel of the BIE, the linear systems in the definite positive case are somehow “diagonal dominant”. The first article seems to be [80] where a one-level strategy in two dimensions for the h-version applied to the weakly singular operator and the hypersingular operator with overlap has been studied. Then, the case of symmetric positive definite operators have been extended to two-level strategy, with and without overlap, for the p-version, h-version and hp-version in 2D [156, 155, 88, 93, 154] and 3D [86, 92, 87, 90, 84, 91, 157]. For an overview on the articles published before 1998, we refer to [147], we also refer to the habilitation thesis [89] for a summary of results on interpolation theory and fractional Sobolev spaces.

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Furthermore, several acceleration procedures can be employed in order to adapt the **method** to the specific symmetries of the structure (translational invariance of some slot, of some posts, etc.) and avoid the computation of duplicated elements in the MoM matrix or in the MM matrix. They have been implemented in the code and used in the simulations discussed in next section.

1 Introduction
During the last 10 years, discontinuous Galerkin (DG) methods have been extensively consid- ered for obtaining approximate solution of Maxwell’s equations, see [CLS04, DFFL10, FLLP05, HW02, HPS04]. Thanks to the discontinuity of the approximation, this kind of methods has many advantages, such as adaptivity to complex geometries through the use of unstructured possibly non-conforming meshes, easily obtained high order accuracy, hp-adaptivity and natural parallelism [HW08]. However, despite these advantages, DG methods have one main drawback particularly sensitive for stationary problems: the number of globally coupled degrees of freedom (DOFs) is much greater than the number of DOFs required by conforming finite element methods for the same accuracy. Consequently, DG methods are expensive in terms of both CPU time and memory consumption, especially for time-harmonic problems [DFLP08]. Hybridization of DG methods [CGL09] is devoted to address this issue while keeping all the advantages of DG methods. The design of such a hybridizable discontinuous Galerkin (HDG) **method** for the dis- cretization of the system of 3d time-harmonic Maxwell’s equations is one of the main objectives of the present work.

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