# Haut PDF Domain decomposition methods for the diffusion equation with low-regularity solution ### Domain decomposition methods for the diffusion equation with low-regularity solution

The outline of the paper is as follows. In section 2, we introduce the notations, geometry and Hilbert spaces to define the problem setting. In particular, we will make use of vector-valued functions with L 2 -jump of normal traces on the interface between subdomains. Then, in section 3, we write the continuous equations and the associated variational formulations of the mixed diffusion equations. We also define the low-regularity case. We next propose an equivalent multi-domain formulation, which fits into the category of domain decomposition methods. The well-posedness of the mixed, multi-domain formulation is studied in section 4 in the continuous case and in section 5 in the discrete case. In the discrete case, we exhibit two abstract algebraic conditions which imply the existence of a discrete inf-sup con- dition. This inf-sup condition ensures well-posedness of the discrete problem, and also convergence when it is uniform. In addition, these algebraic conditions drive the choice of the space of the Lagrange multipliers. We give numerical illustrations in section 6. Finally, we draw some conclusions and give perspectives in section 7.
En savoir plus ### Existence and regularity of solution for a Stochastic Cahn-Hilliard/Allen-Cahn equation with unbounded noise diffusion

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  and I. Gy¨ongy  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.
En savoir plus ### On Domain Decomposition Methods with optimized transmission conditions and cross-points

The issue of cross-points is unavoidable for the development of optimized DDM with enhanced convergence or stability properties. Important mathematical and technical issues are obstacles to the development of these techniques, see [14, 16, 18, 19, 20]. Some partial solutions have been described in [2, 4, 13]. The purely discrete case is considered in [15, 16]. An optimization procedure based on quasi-local operators with convenient regularity is proposed in , but ultimately the problem posed by mesh singularities is not solved due to incompatibility between some function spaces. An important step towards a general solution is currently being investigated on the basis of the recent works of Claeys and Hiptmair  in which a distinct approach is used through the multi-trace formalism and a convenient definition of some function spaces defined on the skeleton of the DDM decomposition. This path has been extended in the PhD thesis of Parolin  and in [9, 8]. However, in this work, we are interested in DDM based on high order transmission conditions such as the ones that are commonly used in numerical DDM softwares, see [3, 20, 22]. To the best of our understanding, none of the works quoted above can provide a sound mathematical treatment of the cross-point issues for such DDM with high order transmission
En savoir plus ### Non overlapping Domain Decomposition Methods for Time Harmonic Wave Problems

Methods for Time Harmonic Wave Problems Xavier Claeys, Francis Collino, Patrick Joly and Emile Parolin Abstract . The domain decomposition method (DDM) initially designed, with the celebrated paper of Schwarz in 1870  as a theoretical tool for partial differential equations (PDEs) has become, since the advent of the computer and parallel com- puting techniques, a major tool for the numerical solution of such PDEs, especially for large scale problems. Time harmonic wave problems offer a large spectrum of applications in various domains (acoustics, electromagnetics, geophysics, ...) and occupy a place of their own, that shines for instance through the existence of a natu- ral (possibly small) length scale for the solutions: the wavelength. Numerical DDMs were first invented for elliptic type equations (e.g. the Laplace equation), and even though the governing equations of wave problems (e.g. the Helmholtz equation) look similar, standard approaches do not work in general.
En savoir plus ### On domain decomposition with space filling curves for the parallel solution of the coupled Maxwell/Vlasov equations

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.
En savoir plus ### Finite element methods with symmetric stabilization for the transient convection-diffusion-reaction equation

1 Introduction The interior penalty method for elliptic and parabolic problems was introduced in 1976 by Douglas and Dupont in the seminal work . In 2004 Burman and Hansbo  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 . The pressure stabilization for Stokes’ problem was considered in  and stability and convergence of the Oseen’s problem at high Reynolds numbers in . 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.
En savoir plus ### Domain decomposition methods and high-order absorbing boundary conditions for the numerical simulation of the time dependent Schrödinger equation with ionization and recombination by intense electric field

the diffusion equation, and more recently by Antoine, Besse , ,  and Szeftel ,  for the linear and nonlinear Schr¨odinger equations by using pseudo- and paradifferential operator techniques. We refer to  for a full literature review on absorbing boundary conditions for TDSE. The goal of the present paper is not to derive new absorbing boundary conditions for the TDSE in order to avoid or limit reflections at domain boundary. We are here interested in the global solution of the laser-molecule TDSE including the ionization and recombination processes. In this goal non-reflecting boundary conditions are not sufficient but will constitute one of the key ingredients, for which the chosen approach is one of the most accurate, and which was developed in . Note that any other ABC could in principle be used  for instance. The other key ingredient is the domain decomposition algorithm to connect the solution in each subdomain. The chosen method is the optimized Schwarz waveform relaxation algorithm (OSWR) , ,  and which was originally proposed in  for the TDSE. In the OSWR method for domain decomposition, transparent boundary conditions are used to derive transmission conditions between subdomains. Using nonlocal transmission conditions was suggested earlier, in particular, in a paper  on domain decomposition for nonlinear elliptic boundary problems.
En savoir plus ### Iterative Methods for Model Reduction by Domain Decomposition

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.
En savoir plus ### Two-Level Domain Decomposition Methods for Highly Heterogeneous Darcy Equations. Connections with Multiscale Methods

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.
En savoir plus ### Reaction-diffusion systems with initial data of low regularity

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. ### Space-Time Domain Decomposition for Mixed Formulations of Diffusion 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
En savoir plus ### Domain decomposition algorithms for the compressible Euler equations

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.
En savoir plus ### Exponentially convergent non overlapping domain decomposition methods for the Helmholtz equation

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.
En savoir plus ### Algebraic Domain Decomposition Methods for Darcy flow in heterogeneous media

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 , 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
En savoir plus ### Domain decomposition methods for non linear problems in fluid dynamics

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. ### Domain decomposition methods for large linearly elliptic three dimensional problems

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. ### Application of fictitious domain method to the solution of the Helmholtz equation in unbounded domain

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. ### Efficient high order and domain decomposition methods for the time-harmonic Maxwell's equations

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.
En savoir plus ### Explicit constants in Harnack inequalities and regularity estimates, with an application to the fast diffusion equation

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 ].
En savoir plus ### Explicit constants in Harnack inequalities and regularity estimates, with an application to the fast diffusion equation

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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