the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

a robin robin preconditioner for advection diffusion equations with discontinuous coefficients abstract we consider an advection diffusion problem with discontinuous viscosity coefficients we apply a substructuring technique and we extend to the resulting schur complement the robin robin preconditioner used for problems with constant viscosity a quasi optimal convergence analysis is performed in the case of uniform convection by means of fourier techniques the variational formulation in order to generalize the preconditioner to an arbitrary number of subdomains is also addressed as well as some numerical tests in 3d

poincare husimi representation of eigenstates in quantum billiards for the representation of eigenstates on a poincare section at the boundary of a billiard different variants have been proposed we compare these poincare husimi functions discuss their properties and based on this select one particularly suited definition for the mean behavior of these poincare husimi functions an asymptotic expression is derived including a uniform approximation we establish the relation between the poincare husimi functions and the husimi function in phase space from which a direct physical interpretation follows using this a quantum ergodicity theorem for the poincare husimi functions in the case of ergodic systems is shown

high performance parallel simulation of structure degradation using non local damage models simulating damage and failure of metallic or composite structures often fails when using standard finite element discretizations and a newton raphson solving procedure the difficulties arise from an uncontrolled mesh dependency caused by damage localization to structural instabilities and to an increase of computational costs the first problem can be overcome by using non local damage constitutive equations together with a specific finite element model the second problem can then be solved with an arc length method which manages instabilities and snap backs they are caused by the loss of strength carrying capacity of the damaged material finally the whole computational scheme is parallelized the main contribution of this paper consists in bringing together the three numerical techniques non local material model piloting and parallel computations the resulting numerical scheme allows to go beyond the overly simplistic cases often encountered in the literature it allows robust industrial simulations with a high number of degrees of freedom both in two and three dimensions and deepened studies involving a large number of simulations

fredholm method for scars a new quasiclassical formula for scars is obtained by using the fredholm method we show that it can be expanded into a formula obtained earlier by agam and fishman the derivation is simple and direct it is also more rigorous and more general than that of agam and fishman it also clarifies the remarkable process of resurgence relating the high order terms based on long orbits to the weyl term whose origin is the zero length orbits

a spatial domain decomposition method for parabolic optimal control problems we present a non overlapping spatial domain decomposition method for the solution of linear quadratic parabolic optimal control problems the spatial domain is decomposed into non overlapping subdomains the original parabolic optimal control problem is decomposed into smaller problems posed on space time cylinder subdomains with auxiliary state and adjoint variables imposed as dirichlet boundary conditions on the space time interface boundary the subdomain problems are coupled through robin transmission conditions this leads to a schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space time interface boundary the schur complement operator is the sum of space time subdomain schur complement operators the application of these subdomain schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem the subdomain schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem we introduce a new family of neumann neumann type preconditioners for the schur complement system including several different coarse grid corrections we compare the numerical performance of our preconditioners with an alternative approach recently introduced by benamou

domain decomposition methods for large linearly elliptic three dimensional problems abstract the idea of solving large problems using domain decomposition technique appears particularly attractive on present day large scale parallel computers but the performance of such techniques used on a parallel computer depends on both the numerical efficiency of the proposed algorithm and the efficiency of its parallel implementation the approach proposed herein splits the computational domain in unstructured subdomains of arbitrary shape and solves for unknown on the interface using the associated trace operator the steklov poincare operator on the continuous level or the schur complement matrix after a finite element discretization and a preconditioned conjugate gradient method this algorithm involves the solution of dirichlet and of neumann problems defined on each subdomain and which can be solved in parallel this method has been implemented on a cray 2 computer using multitasking and on an intel hypercube it was tested on a large scale industrial ill conditioned three dimensional linear elasticity problem which gives a fair indication of its performance in a real life environment in such situations the proposed method appears operational and competitive on both machines compared to standard techniques it yields faster results with far less memory requirements

chiral differential operators formal loop group actions and associated modules abstract chiral differential operators cdos are closely related to string geometry and the quantum theory of 2 dimensional models this paper investigates two topics about cdos on smooth manifolds in the first half we study how a lie group action on a smooth manifold can be lifted to a formal loop group action on an algebra of cdos this turns out to be a condition on the equivariant first pontrjagin class the case of a principal bundle receives particular attention and gives rise to a type of vertex algebras of great interest in the second half we introduce a construction of modules over cdos using the said formal loop group actions and semi infinite cohomology intuitively these modules should have a geometric meaning in terms of formal loop spaces the first example we study leads to a new conceptual construction of an arbitrary algebra of cdos the other example called the spinor module may be useful for a geometric theory of the witten genus

separating the regular and irregular energy levels and their statistics in a hamiltonian system with mixed classical dynamics we look at the high lying eigenstates from the 10001 to the 13000 in the robnik billiard defined as a quadratic conformal map of the unit disk with the shape parameter lambda 0 15 all the 3000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which in comparison with the classical phase space enabled a clear cut classification of energy levels into regular and irregular this is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties we calculate the fractional measure of regular levels as rho 1 0 365 or 0 01 which is in remarkable agreement with the classical estimate rho 1 0 360 or 0 001 this finding confirms the percival s 1973 classification scheme the assumption in berry robnik 1984 theory and the rigorous result by lazutkin 1981 1991 the regular levels obey the poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally brody like statistics with beta 0 286 or 0 001 this is due to the strong localization of irregular eigenstates in the classically chaotic regions therefore in the entire spectrum we see that the berry robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the berry robnik brody distribution

quantum scars of classical closed orbits in phase space the way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x q p through the spectral wigner function w x e this is a sum over wigner functions of eigenstates within a range of energy e in the classical limit w is concentrated on the energy surface and smoothly distributed over it closed orbits provide oscillatory corrections scars for which explicit semiclassical formulae are calculated each scar is a fringe pattern decorating the orbit as x moves off the energy surface the fringes form an airy pattern with spacing of order h as x moves off the closed orbit the fringes form a complex gaussian with spacing h

a scattering approach to the quantization of hamiltonians in two dimensions application to the wedge billiard we extend the scattering approach to quantize general hamiltonian systems in two dimensions and demonstrate it for the wedge billiard the resulting energy levels are given by the zeros of a secular equation the corresponding wavefunctions may also be constructed we verify analytically that this quantization condition is exact in the case of the integrable wedges this method has been applied successfully to computing numerically a few thousand levels for various values of the angle using these data we can check the gutzwiller trace formula and other semiclassical relations involving the scattering matrix and periodic orbits for which we find excellent agreement

affine kac moody algebras and semi infinite flag manifolds we study representations of affine kac moody algebras from a geometric point of view it is shown that wakimoto modules introduced in 18 which are important in conformal field theory correspond to certain sheaves on a semi infinite flag manifold with support on its schhubert cells this manifold is equipped with a remarkable semi infinite structure which is discussed in particular the semi infinite homology of this manifold is computed the cousin grothendieck resolution of an invertible sheaf on a semi infinite flag manifold gives a two sided resolution of an irreducible representation of an affine algebras consisting of wakimoto modules this is just the brst complex as a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent lie algebra

balancing domain decomposition for mixed finite elements the rate of convergence of the balancing domain decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed the balancing domain decomposition method introduced by mandel in 24 is a substructuring method that involves at each iteration the solution of a local problem with dirichlet data a local problem with neumann data and a coarse grid problem to propagate information globally and to insure the consistency of the neumann problems it is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size in both two and three dimensions and for elements of arbitrary order the bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains the key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the h 1 2 norm of an interplant of the boundary data computational results from a message passing parallel implementation on an intel delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed up for up to 64 processors

overlapping schwarz preconditioners for spectral element discretizations of convection diffusion problems the classical overlapping schwarz algorithm is here extended to stabilized spectral element discretizations of convection diffusion problems the algorithm solves iteratively the resulting non symmetric system of linear equations by a preconditioned gmres method the preconditioner is built from local convection diffusion solvers on overlapping subdomains and from a coarse convection diffusion solver on a coarse mesh defined by the subdomain boundaries several numerical experiments on test problems in the plane indicate that this algorithm retains the fast convergence rate and optimal scalability properties of classical overlapping methods for diffusion dominated problems fast convergence is also obtained for convection dominated problems without closed streamlines and with a moderate number of subdomains copyright 2001 john wiley sons ltd

domain decomposition methods for advection dominated linear quadratic elliptic optimal control problems abstract we present an optimization level domain decomposition dd preconditioner for the solution of advection dominated elliptic linear quadratic optimal control problems which arise in many science and engineering applications the dd preconditioner is based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through robin transmission conditions for the states and the adjoints the parameters in the robin transmission condition depend on the advection this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with robin boundary conditions for the state numerical tests for problems with distributed and with boundary control show that the dependence of the preconditioners on mesh size and subdomain size is comparable to its counterpart applied to a single advection dominated equation these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains three iterative domain decomposition methods are considered simultaneous updates on all subdomains additive schwarz method flow directed sweeps and double sweeps by using some techniques of formal language theory we obtain a unique criterion of convergence for the three methods the convergence rate is a function of the criterion and depends on the algorithm

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

two level domain decomposition preconditioning for the p version finite element method in three dimensions this paper presents a flexible approach to domain decomposition preconditioning for problems expressed in terms of local stiffness matrices of subdomains evaluation of the preconditioner requires fully parallelizable local computations and the solution of a global auxiliary problem with few variables per subdomain convergence factors can be bounded using properties of the local stiffness matrices only the method is applied to the p version finite element method for three dimensional elasticity we treat each element as a subdomain and compute explicit convergence bounds

feti domain decomposition methods for scalar advection diffusion problems abstract in this paper we show that iterative substructuring methods of finite element tearing and interconnecting type can be successfully employed for the solution of linear systems arising from the finite element approximation of scalar advection diffusion problems using similar ideas as those of a recently developed neumann neumann method we propose a one level algorithm and a class of two level algorithms obtained by suitably modifying the local problems on the subdomains we present some numerical results for some significant test cases our methods appear to be optimal for flows without closed streamlines and possibly very small values of the viscosity they also show very good performances for rotating flows and moderate reynolds numbers therefore the algorithms proposed appear to be well suited for many convection dominated problems of practical interest

statistical analysis of scars in stadium billiard in this paper by using our improved plane wave decomposition method we study the scars in the eigenfunctions of the stadium billiard from a very low state to as high as about the 1 millionth state in the systematic searching for scars of various types we have used the approximate criterion based on the quantization of the classical action along the unstable periodic orbit supporting the scar we have analysed the profile of the integrated probability density along the orbit we found that the maximal integrated intensity of different types of scars scales in different way with the which confirms qualitatively and quantitatively the existing theories of scars such as that of bogomolny and robnik

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

quantum phase space structures in classically mixed systems a scattering approach the structure of quantum wavefunctions is discussed for a billiard system with mixed classical dynamics using a scattering formalism we introduce a husimi like distribution for s matrix eigenvectors at arbitrary wavenumbers k not necessarily eigenenergies of the system this constitutes probability density plots on the poincare surface of section at any k many eigenvectors are structured by classical objects such as invariant tori the intermediate layer between regular and chaotic motion and unstable periodic orbits scars we find that eigenvector structure is stable over significant energy ranges while the associated eigenphases increase linearly with k at a rate determined by the underlying classical objects additionally we observe that in the presence of time reversal symmetry eigenvectors scarred by non self retracing periodic orbits form doublets which are closely spaced in eigenphase our results provide a new perspective on the quantization of structured states quantization is determined by the linearly increasing eigenphases and occurs whenever an eigenphase is equal to while structure is encoded in the eigenvectors it is present at all energies and varies only slowly this allows one to predict many quantized eigenfunctions over an entire energy range by studying the eigenparameters of at a single intermediate energy and identifying the underlying classical objects

gerbes of chiral differential operators ii vertex algebroids introduction 605 0 recollections and notation 612 1 vertex algebroids 615 2 from vertex algebras to vertex algebroids 622 3 category of vertex algebroids 628 4 cofibered structure 633 5 computation of a cocycle 637 6 characteristic classes of vector bundles 643 7 torseurs and gerbes 647 8 gerbes of vertex algebroids 653 9 chiral differential operators on curves and sheaves of virasoro algebras 657 10 enveloping algebra of a vertex algebroid 664 11 calabi yau structures 668 12 virasoro and calabi yau structures 671 references 680

deriving a new domain decomposition method for the stokes equations using the smith factorization in this paper the smith factorization is used systematically to derive a new domain decomposition method for the stokes problem in two dimensions the key idea is the transformation of the stokes problem into a scalar bi harmonic problem we show how a proposed domain decomposition method for the bi harmonic problem leads to a domain decomposition method for the stokes equations which inherits the convergence behavior of the scalar problem thus it is sufficient to study the convergence of the scalar algorithm the same procedure can also be applied to the three dimensional stokes problem as transmission conditions for the resulting domain decomposition method of the stokes problem we obtain natural boundary conditions therefore it can be implemented easily a fourier analysis and some numerical experiments show very fast convergence of the proposed algorithm our algorithm shows a more robust behavior than neumann neumann or feti type methods

a domain decomposition strategy for nonclassical frictional multi contact problems in this paper we present a numerical strategy to be solve large scale frictional contact problems by domain decomposition methods which are adapted to parallel computers the motivation is given by the study of the mechanical behavior of rolling shutters composed by many hinged slats the numerical treatment of such nonclassical contact problems leads to very large strongly nonlinear nonsymmetric and ill conditioned systems domain decomposition methods are a good alternative to overcome the difficulties of classical sequential solutions we present a nonlinear strategy adapted to problems called multi contact problems

semi infinite induction and wakimoto modules the purpose of this paper is to suggest the construction and study properties of semi infinite induction which relates to semi infinite cohomology the same way induction relates to homology and coinduction to cohomology we prove a version of the shapiro lemma showing that the semi infinite cohomology of a module is isomorphic to that of the semi infinitely induced module a practical outcome of our construction is a simple construction of the wakimoto modules highest weight modules used in double sided bgg resolutions of irreducible modules

an iteration by subdomain overlapping dirichlet robin domain decomposition method for advection diffusion problems we present a new overlapping dirichlet robin domain decomposition method the method uses dirichlet and robin transmission conditions on the interfaces of an overlapping partitioning of the computational domain we derive interface equations to study the convergence of the method and show its properties through four numerical examples the mathematical framework is general and can be applied to derive overlapping versions of all the classical nonoverlapping methods

vertex algebras and mirror symmetry mirror symmetry for calabi yau hypersurfaces in toric varieties is by now well established however previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror we are able to calculate explicitly vertex algebras that correspond to holomorphic parts of a and b models of calabi yau hypersurfaces and complete intersections in toric varieties we establish the relation between these vertex algebras for mirror calabi yau manifolds this should eventually allow us to rewrite the whole story of toric mirror symmetry in the language of sheaves of vertex algebras our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra as well as some basic results of the theory of vertex algebras ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties

balancing neumann neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity balancing neumann neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients discretized with mixed finite or spectral elements with discontinuous pressures these domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions a polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case parallel and serial numerical experiments confirm the theoretical results indicate that they still hold for systems with discontinuous coefficients and show that our algorithm is scalable parallel and robust with respect to material heterogeneities the results on heterogeneous general problems are also supported in part by our theory

positive definite balancing neumann neumann preconditioners for nearly incompressible elasticity in this paper a positive definite balancing neumann neumann bnn solver for the linear elasticity system is constructed and analyzed the solver implicitly eliminates the interior degrees of freedom in each subdomain and solves iteratively the resulting schur complement involving only interface displacements using a bnn preconditioner based on the solution of a coarse elasticity problem and local elasticity problems with natural and essential boundary conditions while the schur complement becomes increasingly ill conditioned as the materials becomes almost incompressible the bnn preconditioned operator remains well conditioned the main theoretical result of the paper shows that the proposed bnn method is scalable and quasi optimal in the constant coefficient case this bound holds for material parameters arbitrarily close to the incompressible limit while this result is due to an underlying mixed formulation of the problem both the interface problem and the preconditioner are positive definite numerical results in two and three dimensions confirm these good convergence properties and the robustness of the methods with respect to the almost incompressibility of the material

a numerical study on neumann neumann and feti methods for hp approximations on geometrically refined boundary layer meshes in two dimensions abstract we present extensive numerical tests showing the performance and robustness of certain balancing neumann neumann and one level feti methods for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in two dimensions the numerical results confirm the theoretical results derived in a toselli x vasseur technical report 02 15 seminar fur angewandte mathematik eth zurich september 2002 the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients in addition they only grow polylogarithmically with the polynomial degree as in the case of p approximations on shape regular meshes

smoothed wave functions of chaotic quantum systems abstract it is shown that wave functions of quantum systems as 0 have an extra density near unstable periodic trajectories of the classical problem the averaged wave function square is represented as the sum over a finite number of periodic trajectories the contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory the results are compared with the numerical calculations of the wave functions for the stadium billiard

chiral differential operators on supermanifolds the first part of this paper provides a new description of chiral differential operators cdos in terms of global geometric quantities the main result is a recipe to define all sheaves of cdos on a smooth cs manifold its ingredients consist of an affine connection and an even 3 form that trivializes p1 with fixed two suitable 3 forms define isomorphic sheaves of cdos if and only if their difference is exact moreover conformal structures are in one to one correspondence with even 1 forms that trivialize c1 applying our work in the first part we then construct what may be called chiral dolbeault complexes of a complex manifold m and analyze conditions under which these differential vertex superalgebras admit compatible conformal structures or extra gradings by fermion numbers when m is compact their cohomology computes in various cases the witten genus the two variable elliptic genus and a spinc version of the witten genus this part contains some new results as well as provides a geometric formulation of certain known facts from the study of holomorphic cdos and models

statistical properties of high lying chaotic eigenstates we study the statistical properties of the high lying chaotic eigenstates 200000 and above which are deep within the semiclassical regime the system we are analysing is the billiard system inside the region defined by the quadratic complex conformal map of the unit disk as introduced by robnik 1983 we are using heller s method of plane wave decomposition of the numerical eigenfunctions and perform extensive statistical analysis with the following conclusions i the local average probability density is in excellent agreement with the microcanonical assumption and all statistical properties are also in excellent agreement with the gaussian random model ii the autocorrelation function is found to be strongly direction dependent and only after averaging over all directions agrees well with berry s 1977 prediction iii although the scars of unstable classical periodic orbits in such an ergodic regime are expected to exist so far we have not found any around the 200000th state other than a scar like feature resembling the whispering gallery modes

an efficient fredholm method for the calculation of highly excited states of billiards a numerically efficient fredholm formulation of the billiard problem is presented the standard solution in the framework of the boundary integral method in terms of a search for roots of a secular determinant is reviewed first we next reformulate the singularity condition in terms of a flow in the space of an auxiliary one parameter family of eigenproblems and argue that the eigenvalues and eigenfunctions are analytic functions within a certain domain based on this analytic behavior we present a numerical algorithm to compute a range of billiard eigenvalues and associated eigenvectors by only two diagonalizations

balancing neumann neumann methods for the cardiac bidomain model balancing neumann neumann preconditioners are constructed analyzed and numerically studied for the cardiac bidomain model in three dimensions this reaction diffusion system is discretized by low order finite elements in space and implicit explicit methods in time yielding very ill conditioned linear systems that must be solved at each time step the proposed algorithm is based on decomposing the domain into nonoverlapping subdomains and on solving iteratively the bidomain schur complement obtained by implicitly eliminating the degrees of freedom interior to each subdomain the iteration is preconditioned by a balancing neumann neumann method employing local neumann solves on each subdomain and a coarse bidomain solve a novel approach for the estimation of the average operator of the nonoverlapping decomposition provides a framework for designing coarse spaces for balancing neumann neumann methods the theoretical estimates obtained show that the proposed method is scalable quasi optimal and robust with respect to possible coefficient discontinuities of the bidomain operator the results of extensive parallel numerical tests in three dimensions confirm the convergence rates predicted by the theory

a domain decomposition preconditioner for an advection diffusion problem abstract we propose a generalization of the neumann neumann algorithm for advection diffusion problems the neumann conditions are replaced by suitable robin conditions the method is first introduced and analysed in the case where the domain is partitioned into vertical strips then generalized to an arbitrary decomposition a coarse space procedure is proposed finally numerical results for bidimensional and tridimensional tests are given

wavefunctions expectation values and scars on poincare sections a scattering approach using the scattering approach to quantization we define an appropriate representation of the billiard wavefunction on the scattering poincar section the expectation values of smooth operators in terms of these poincar section wavefunctions are expressed as sums over periodic orbits a special operator is used to define scars on the section and the relation to scars in configuration space is discussed and demonstrated numerically

bddc preconditioning for high order galerkin least squares methods using inexact solvers a high order galerkin least squares gls finite element discretization is combined with a balancing domain decomposition by constraints bddc preconditioner and inexact local solvers to provide an efficient solution technique for large scale convection dominated problems the algorithm is applied to the linear system arising from the discretization of the two dimensional advection diffusion equation and euler equations for compressible inviscid flow a robin robin interface condition is extended to the euler equations using entropy symmetrized variables the bddc method maintains scalability for the high order discretization of the diffusion dominated flows and achieves low iteration count in the advection dominated regime the bddc method based on inexact local solvers with incomplete factorization and p 1 coarse correction maintains the performance of the exact counterpart for the wide range of the peclet numbers considered