Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods

We present numerical results concerning the solution of the time-harmonic Maxwell equations discretized by discontinuous Galerkin methods. In particular, a numerical study of the convergence, which compares different strategies proposed in the literature for the elliptic Maxwell equations, is performed in the two-dimensional case.     

Antioxidant activity of rosemary extracts in solution and embedded in polymeric systems

The rosemary extract was encapsulated in polyethylene or in covalently-based network gels. The covalent gels were obtained by the reaction of isocyanate end-capped polyethylene glycol (PEG) with β-cyclodextrin or glycerol. The 2,2-diphenyl-1-picrylhydrazyl (DPPH) assay was used to evaluate the antioxidant activity (AA) of rosemary extract entrapped in polymeric structures and in ethanol or water solutions. The AA of the rosemary extract was determined using a DPPH radical for samples prepared in ethanol, and a water-soluble derivative, the sulphonated DPPH radical (DPPH-SO₃Na), for the rosemary extract in water. Formulation of the rosemary extract in polymeric gels ensures a rapid release which determines the AA values similar to those in solution.     

New constructions of domain decomposition methods for systems of PDEs

We propose new domain decomposition methods for systems of partial differential equations in two and three dimensions. The algorithms are derived with the help of the Smith factorization. This could also be validated by numerical experiments. To cite this article: V. Dolean et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

4-(Azulen-1-yl) six-membered heteroaromatics substituted in 2- and 6- positions with 2-(2-furyl)vinyl, 2-(2-thienyl)vinyl or 2-(3-thienyl)vinyl moieties

The synthesis of pyrylium and pyridinium salts and pyridines with azulene-1-yl moieties in position 4 and two 2-heteroarylvinyl groups in positions 2 and 6 was accomplished. The pyrylium salts were obtained starting from pyranones and pyridines could be prepared from these salts by treating them with ammonium acetate. The general procedures for the synthesis of pyridinium salts, which occur with good results in less delocalized electronic systems, do not take place when applied to the above obtained pyrylium salts. Therefore, as starting material 4-(azulen-1-yl)-1-(n-butyl)-2,6-dimethylpyridinium perchlorate was used, which was condensed with heteroarylcarboxaldehydes. These compounds were completely characterized and some of their spectra were discussed. Their interaction with some metal ions was revealed, observing an affinity better than in the case of simple azulenepyridines. In the last part of the paper are presented redox potentials for several pyrylium salts and pyridines in comparison with those of the nonvinylogated derivatives.     

A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods

We present here a domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by a discontinuous Galerkin method. In order to allow the treatment of irregularly shaped geometries, the discontinuous Galerkin method is formulated on unstructured tetrahedral meshes. The domain decomposition strategy takes the form of a Schwarz-type algorithm where a continuity condition on the incoming characteristic variables is imposed at the interfaces between neighboring subdomains. A multifrontal sparse direct solver is used at the subdomain level. The resulting domain decomposition strategy can be viewed as a hybrid iterative/direct solution method for the large, sparse and complex coefficients algebraic system resulting from the discretization of the time-harmonic Maxwell equations by a discontinuous Galerkin method.     

A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes

We report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two‐dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi‐discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo‐time step requires the solution of a sparse linear system for the flow variables. In this study, a non‐overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson‐type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical (normal) fluxes. Copyright © 2001 John Wiley & Sons, Ltd.     

Solving generalized eigenvalue problems on the interfaces to build a robust two-level FETI method

FETI is a very popular method, which has proved to be extremely efficient on many large-scale industrial problems. One drawback is that it performs best when the decomposition of the global problem is closely related to the parameters in equations. This is somewhat confirmed by the fact that the theoretical analysis goes through only if some assumptions on the coefficients are satisfied. We propose here to build a coarse space for which the convergence rate of the two-level method is guaranteed regardless of any additional assumptions. We do this by identifying the problematic modes using generalized eigenvalue problems.     

A robust two-level domain decomposition preconditioner for systems of PDEs

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial differential equations, especially for systems. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems which isolate the terms responsible for slow convergence. We give a general theoretical result and then some numerical examples on a heterogeneous elasticity problem.     

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.