1.3. Main results. As a first step **for** a rigorous mathematical analysis of **the** stochastic model, in Section 2, we will prove existence and uniqueness of a **solution** to (1.1) when **the** initial condition u 0 belongs to L q (D) **for** q ∈ [3, ∞) if d = 1, 2 and q ∈ [6, ∞) if d = 3. Section 3 describes some possible general assumptions on **the** **domain** D which would lead to **the** same result and presents **the** stochastic Cahn-Hilliard **equation** as a special case of a Cahn-Hilliard/Allen-Cahn stochastic model. Note that **the** approach used in this paper to solve this non linear SPDE **with** a polynomial growth is similar to that developed by J.B. Walsh [22] and I. Gy¨ongy [16] **for** **the** stochastic heat **equation** and related SPDEs. Note that unlike these references, **the** smoothing effect of **the** bi-Laplace operator enables us to deal **with** a stochastic perturbation driven by a space-time white noise in dimension 1 up to 3.

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2.7.3 Influence of **the** edge-cut **for** a particle simulation
A particle simulation benefits from a DD **with** a small edge-cut at least at two steps. As al- ready discussed in 2.5.2 there is firstly communication of boundary data during **the** **solution** of **the** Maxwell **equation**. **The** edge-cut provides an estimation **for** **the** transfer volume hence it is beneficial to keep **the** edge-cut **low**. Secondly a small edge-cut is also beneficial **for** **the** particle push phase, more precisely **the** amount of particle migrations that are necessary during **the** simulation. Due to some Courant-Friedrichs-Levy condition (CFL)-condition **for** **the** **solution** of **the** Maxwell equations **the** time step in particle simulations is limited which leads to a limited distance a particle can travel during one particle push phase. Further, **with** decreasing edge-cut, **the** probability that **the** neighborhood of a particle’s surround- ing tetrahedron contains tetrahedra from other partitions decreases. **For** this reason **the** likelihood that a particle remains on a certain processor during one particle push phase increases **with** decreasing edge-cut. Less particle migration keeps a **decomposition** balanced and minimizes **the** communication required **for** distributing particles.

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1 Introduction
**The** interior penalty method **for** elliptic and parabolic problems was introduced in 1976 by Douglas and Dupont in **the** seminal work [8]. In 2004 Burman and Hansbo [5] proved that **the** method was robust at high Peclet numbers and en- joyed **the** same stability properties as **the** Streamline-**Diffusion** (SD) method. A number of extensions to various problems in fluid mechanics were then proposed by Burman and co-workers. An extension to non-conforming approximation spaces was proposed in [3]. **The** pressure stabilization **for** Stokes’ problem was considered in [6] and stability and convergence of **the** Oseen’s problem at high Reynolds numbers in [4]. **The** method has several advantages compared to **the** SD-method, mainly thanks to **the** fact that **the** stabilizing term does not couple to **the** **low** order residual and is therefore independent of both time deriva- tives and source terms. Hence, space and time discretization commute and **the** method can be combined **with** any type of time discretisation and nodal quadrature leads to a diagonal matrix contribution from stiff source terms. An- other important feature is that **the** stabilization parameter is independent of **the** **diffusion** parameter and more generally has less dependence of **the** problem parameters than **the** SD method, since consistency of **the** SD-method depends on **the** residual of **the** differential **equation**, whereas **the** CIP method is weakly consistent, depending only on **the** **regularity** of **the** exact **solution**.

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Abstract: We propose a method to reduce **the** computational effort to solve a partial differential **equation** on a given **domain**. **The** main idea is to split **the** **domain** of interest in two subdomains, and to use different approximation **methods** in each of **the** two subdomains. In particular, in one subdomain we discretize **the** governing equations by a canonical scheme, whereas in **the** other one we solve a reduced order model of **the** original problem. Different approaches to couple **the** **low**- order model to **the** usual discretization are presented. **The** effectiveness of these approaches is tested on numerical examples pertinent to non-linear model problems including **the** Laplace **equation** **with** non-linear boundary conditions and **the** compressible Euler equations.

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Multiphase, compositional porous media ﬂow models, used in reservoir simulations or basin modeling, lead to **the** **solution** of complex non linear systems of Partial Differential Equations (PDE). These PDE are typically discretized using a cell-centered ﬁnite volume scheme and a fully implicit Euler integration in time in order to allow **for** large time steps. After Newton type lineari- zation, one ends up **with** **the** **solution** of a linear system at each Newton iteration which represents up to 90 per- cents of **the** total simulation elapsed time. **The** corre- sponding pressure block matrix is related to **the** discretization of a Darcy **equation** **with** high contrasts and anisotropy in **the** coefﬁcients. We focus on overlap- ping Schwarz type **methods** on parallel computers and on multiscale **methods**.

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basic estimates initiated by M. Pierre and collaborators, who have introduced **methods** to prove L 2
a priori estimates **for** **the** **solution**.
Here, we establish such a key estimate **with** initial data in L 1 while **the** usual theory uses L 2 .
This allows us to greatly simplify **the** proof of some results. We also establish new existence results of semilinearity which are super-quadratic as they occur in complex chemical reactions. Our method can be extended to semi-linear porous medium equations.

5. Gander, M.J., Halpern, L., Nataf, F.: Optimized Schwarz waveform relaxation method **for** **the** one dimensional wave **equation**. SIAM J. Numer. Anal. 41(5) (2003)
6. Gander, M.J., Japhet, C., Maday, Y., Nataf, F.: A new cement to glue non-conforming grids **with** Robin interface conditions: **The** finite element case. In: R. Kornhuber, R.H.W. Hoppe, J. P´eriaux, O. Pironneau, O.B. Widlund, J. Xu (eds.) Proceedings of **the** 5th International Con- ference on **Domain** **Decomposition** **Methods**, vol. 40, pp. 259–266. Springer LNCSE (2005) 7. Halpern, L., Japhet, C., Omnes, P.: Nonconforming in time **domain** **decomposition** method **for**

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1 Introduction
When solving **the** compressible Euler equations by an implicit scheme **the** nonlinear system is usually solved by Newton’s method. At each step of this method we have to solve a linear system which is non-symmetric and very ill conditioned. **The** necessity of a **domain** **decomposition** became more and more obvious. In a previous paper [DLN04] we formulated a Schwarz algorithm (interface iteration which relies on **the** successive solving of **the** local decomposed problems and **the** transmission of **the** result at **the** interface) involving transmission conditions that are derived naturally from a weak formulation of **the** underlying boundary value problem (first formulated in [QS96]). As far as these algorithms are con- cerned, when dealing **with** supersonic flows, whatever **the** space dimension is, imposing **the** appropriate characteristic variables as interface conditions leads to a convergence of **the** algorithm which is optimal **with** regards to **the** number of subdomains. **The** only case of interest remains **the** subsonic one where this property is lost except in **the** one dimensional case. We recall briefly these results in order to introduce more general and performant **methods** such as **the** optimized interface conditions and **the** preconditioning **methods**. **The** former were widely studied and analyzed **for** scalar problems such as elliptic equations in [Lio90, EZ98], **for** **the** Helmholtz **equation** in [BD97, CN98] convection-**diffusion** problems in [JNR01]. **For** time dependent problems and local times steps, see **for** instance [GHN01]. **The** preconditioning **methods** have also known a wide developpement in **the** last decade. **The** Neumann-Neumann algorithms **for** sym- metric second order problems [RT91] has been **the** subject of numerous works, see [TW04] and references therein. An extension of these algorithms to non-symmetric scalar problems (**the** so called Robin-Robin algorithms) has been done in [ATNV00, GGTN04] **for** advection-**diffusion** problems.

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to **the** **regularity** of 𝜒.
This allowed us to compute **the** function 𝜏 𝑟 * (𝐿, 𝑘). In Figure 11 , we have represented **the** curves 𝐿/𝜆 → 𝜏 𝑟 * (𝐿, 𝑘) where 𝜆 = 2𝜋/𝑘 is **the** wavelength, **for** two values of 𝑘: 𝑘 = 2𝜋 and 𝑘 = 6𝜋.
On each curve, we observe that **the** optimized rate does not depend on 𝐿 as long as 𝐿/𝜆 is sufficiently large and exceeds 1 4 . Of course, when 𝐿 goes to zero, this convergence rate tends to 1, which is expected since **the** operator is no longer non local. This observation seems to be relatively independent from **the** frequency. From **the** practical point of view, this means that one could get **the** effect of our non local operator (exponential convergence) at **the** cost of a local operator. Indeed, **with** standard finite element **methods**, one is used to fix **the** number of grid points per wavelength. Then choosing a truncation length proportional to **the** wavelength, **the** number of grid points in **the** support of **the** 𝑦 → 𝜒(|𝑥 − 𝑦|/𝐿) is approximately constant so that **the** sparsity of **the** matrix corresponding to **the** non local operator would be in practice independent on **the** wavelength.

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by defining custom weights **for** all edges in graph. When we relate graph partition mecha- nism i.e., cutting graph edges, to highly anisotropic problem, **the** way how we should define values of **the** edges weights comes naturally. In order to avoid partition **with** interfaces of **the** subdomains going along a direction of anisotropy we need to associate edges parallel to this direction **with** big weight. Thus from **the** minimisation process point of view, it will be a big “cost” **for** partitioner to cut them, in contrary to edges which are oriented perpen- dicularly to anisotropy direction which we associate **with** small weights in such way that it will be “cheap” to cut them. However, we do not need to know **the** geometrical orientation of edges to compute **for** them suitable weights. Some simple calculation inherited from al- gebraic multi grids techniques [56], express **the** desired properties described above. Thus, since number of edges of **the** adjacency graph is equal to **the** number of non-zeros in under- lying sparse matrix, we can easily compute edge weight using values of underlying matrix via following formula

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Résumé. Les équations de Maxwell en régime harmonique comportent plusieurs difficultés lorsque la fréquence est élevée. On peut notamment citer le fait que leur formulation variationnelle n’est pas définie positive et l’effet de pollution qui oblige à utiliser des maillages très fins, ce qui rend problématique la construction de solveurs itératifs efficaces. Ici nous proposons une stratégie de **solution** précise et rapide, qui associe une discrétisation par des éléments finis d’ordre élevé à des préconditionneurs de type décomposition de domaine. Les éléments finis d’ordre élevé permettent, pour une précision donnée, de réduire considérablement le nombre d’inconnues du sytème linéaire à résoudre. Ensuite, des méthodes de décomposition de domaine sont employées comme préconditionneurs du sytème linéaire pour le solveur itératif : le problème défini sur le domaine global est décomposé en des problèmes plus petits sur des sous-domaines, qui peuvent être résolus en parallèle et avec des solveurs directs robustes. Cependant la conception, l’implémentation et l’analyse des deux méthodes sont assez difficiles pour les équations de Maxwell. Les éléments finis adaptés à l’approximation du champ électrique sont les éléments finis H(rot)-conformes ou d’arête. Ici nous revisitons les degrés de liberté classiques définis par Nédélec, afin d’obtenir une expression plus pratique par rapport aux fonctions de base d’ordre élevé choisies. De plus, nous proposons une technique pour restaurer la dualité entre les fonctions de base et les degrés de liberté. Nous décrivons explicitement une stratégie d’implémentation qui a été appliquée dans le langage spécialisé et open source FreeFem++. Dans une deuxième partie, nous nous concentrons sur les techniques de préconditionnement du système linéaire résultant de la discrétisation par éléments finis. Nous commençons par la validation numérique d’un préconditionneur à un niveau, de type Schwarz avec recouvrement, avec des conditions de transmission d’impédance entre les sous-domaines. Ensuite, nous étudions comment des préconditionneurs à deux niveaux, analysés récemment pour l’équation de Helmholtz, se comportent pour les équations de Maxwell, des points de vue théorique et numérique. Nous appliquons ces méthodes à un problème à grande échelle qui découle de la modélisation d’un système d’imagerie micro-onde, pour la détection et le suivi des accidents vasculaires cérébraux. En effet, la précision et la vitesse de calcul sont essentielles dans cette application.

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This document is divided into two Parts. Part I is devoted to **the** explicit computa- tion of **the** constant in Moser’s Harnack inequality based on **the** method of [ 25 , 26 ]. As far as we know, no such expression of **the** constant has yet been published. Part II is devoted to fully explicit and constructive estimates which are needed **for** proving **the** stability in some Gagliardo-Nirenberg inequalities in [ 9 ]. **For** a comprehensive introduction to stability issues and a review of **the** literature, **the** reader is invited to refer to [ 9 , Section 1]. Boxed inequalities are used to quote results of [ 9 ].

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v . (12)
Here a weak **solution** is defined as in [ 26 , p. 728], [ 24 , Chapter 3] or [ 3 ]. **The** Harnack inequality of Theorem 1 goes back to J. Moser [ 25 , 26 ]. **The** dependence of **the** constant on **the** ellipticity constants λ 0 and λ 1 was not clear before **the** paper of J. Moser [ 26 ], where he shows that such a dependence is optimal by providing an explicit example, [ 26 , p. 729]. **The** fact that h only depends on **the** dimension d is also pointed out by C.E. Gutierrez and R.L. Wheeden in [ 22 ] after **the** statement of their Harnack inequalities, [ 22 , Theorem A]. However, to our knowledge, a complete constructive proof and an expression like ( 11 ) was still missing. We do not claim any originality concerning **the** strategy but provide **for** **the** first time an explicit expression **for** **the** constant h.

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