a homographic best approximation problem with application to optimized schwarz waveform relaxation we present and study a homographic best approximation problem which arises in the analysis of waveform relaxation algorithms with optimized transmission conditions its solution characterizes in each class of transmission conditions the one with the best performance of the associated waveform relaxation algorithm we present the particular class of first order transmission conditions in detail and show that the new waveform relaxation algorithms are well posed and converge much faster than the classical one the number of iterations to reach a certain accuracy can be orders of magnitudes smaller we illustrate our analysis with numerical experiments

subnormal and quasinormal toeplitz operators with matrix valued rational symbols abstract in this paper we deal with the subnormality and the quasinormality of toeplitz operators with matrix valued rational symbols in particular in view of halmos s problem 5 we focus on the question which subnormal toeplitz operators are normal or analytic we first prove let l m n be a matrix valued rational function having a matrix pole i e there exists d for which ker h z h c n 2 where h denotes the hankel operator with symbol if i t is hyponormal ii ker t t is invariant for t then t is normal hence in particular if t is subnormal then t is normal next we show that every pure quasinormal toeplitz operator with a matrix valued rational symbol is unitarily equivalent to an analytic toeplitz operator

a residual based a posteriori error estimator for the stokes darcy coupled problem in this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow the flows are governed by the stokes and darcy equations respectively and the transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law the finite element subspaces consider bernardi raugel and raviart thomas elements for the velocities piecewise constants for the pressures and continuous piecewise linear elements for a lagrange multiplier defined on the interface we derive a reliable and efficient residual based a posteriori error estimator for this coupled problem the proof of reliability makes use of suitable auxiliary problems diverse continuous inf sup conditions satisfied by the bilinear forms involved and local approximation properties of the clement interpolant and raviart thomas operator on the other hand helmholtz decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are the main tools for proving the efficiency of the estimator up to minor modifications our analysis can be extended to other finite element subspaces yielding a stable galerkin scheme

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

on a class of jointly hyponormal toeplitz operators we characterize when a pair of toeplitz operators t t t is jointly hyponormal under various assumptions for example is analytic or is a trigonometric polynomial or is analytic a typical characterization states that t t t is jointly hyponormal if and only if an algebraic relation of and holds and the single toeplitz operator t is hyponormal where is a combination of and more general results for an n tuple of toeplitz operators are also obtained

a convergence theory for an overlapping schwarz algorithm using discontinuous iterates a new type of overlapping schwarz methods using discontinuous iterates is constructed by modifying the classical overlapping schwarz algorithm this new algorithm allows for discontinuous iterates across the artificial interface for poisson s equation this algorithm can be considered as an overlapping version of lions robin iteration method for which little is known concerning the rate of convergence since overlap improves the performance of the classical algorithms considerably the existence of a uniform convergence factor is the fundamental question for our algorithm a new theory using lagrange multipliers is developed and conditions are found for the existence of an almost uniform convergence factor for the dual variables which implies rapid convergence of the primal variables in the two overlapping subdomain case our result also shows a relation between the boundary conditions of the given problem and the artificial interface condition numerical results for the general case with cross points are also presented they indicate possible extensions of our results to this more general case

non conformal domain decomposition method with second order transmission conditions for time harmonic electromagnetics non overlapping domain decomposition dd methods provide efficient algorithms for solving time harmonic maxwell equations it has been shown that the convergence of dd algorithms can be improved significantly by using high order transmission conditions in this paper we extend a newly developed second order transmission condition sotc which involves two second order transverse derivatives to facilitate fast convergence in the non conformal dd algorithms however the non conformal nature of the dd methods introduces an additional technical difficulty which results in poor convergence in many real life applications to mitigate the difficulty a corner edge penalty method is proposed and implemented in conjunction with the sotc to obtain truly robust solver performance numerical results verify the analysis and demonstrate the effectiveness of the proposed methods on a few model problems finally drastically improved convergence compared to the conventional robin transmission condition was observed for an electrically large problem of practical interest

hyponormal toeplitz operators and extremal problems of hardy spaces the symbols of hyponormal toeplitz operators are completely described and those are also studied being related with the extremal problems of hardy spaces moreover we discuss halmos s question about a subnormal toeplitz operator when the self commutator is finite rank

optimized schwarz waveform relaxation for the primitive equations of the ocean in this article we are interested in the derivation of efficient domain decomposition methods for the viscous primitive equations of the ocean we consider the rotating three dimensional incompressible hydrostatic navier stokes equations with free surface performing an asymptotic analysis of the system in the regime of small rossby numbers we compute an approximate dirichlet to neumann operator and build an optimized schwarz waveform relaxation algorithm we establish that the algorithm is well defined and provide numerical evidence of the convergence of the method

optimized schwarz methods with robin transmission conditions for parabolic problems in this paper optimized schwarz methods with robin transmission conditions are considered for solving the time dependent linear algebraic systems resulting from finite element approximations for parabolic problems we use both analysis and numerical experiments to investigate the convergence behavior particularly we check the influence of the time step size on the convergence rate we get the estimate of the upper bound of convergence rate 1 o min h 1 2 h 1 2 h 1 2 h 3 2 tau 1 where h is the mesh size h is the size of subdomains and tau is the time step size our numerical results show that the convergence rate of this method is fast we also search for the optimal parameter lambda by experiments and find the rule of its distribution

a parallel robin robin domain decomposition method for the stokes darcy system we propose a new parallel robin robin domain decomposition method for the coupled stokes darcy system with beavers joseph saffman jones interface boundary condition in particular we prove that with an appropriate choice of parameters the scheme converges geometrically independent of the mesh size

hyponormality of toeplitz operators with generalized circulant symbols in this paper we obtain a necessary and sufficient condition for toeplitz operators with generalized circulant symbols to be hyponormal

siegel disks with smooth boundaries we show the existence of angles r z such that the quadratic polynomial p z e2i z z2 has a siegel disk with c smooth boundary this result was first announced by r perez marco in 1993

a rapidly converging domain decomposition method for the helmholtz equation a new domain decomposition method is introduced for the heterogeneous 2 d and 3 d helmholtz equations transmission conditions based on the perfectly matched layer pml are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces we focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes in combination with a certain ordering for solving the subdomain problems and a gmres outer iteration when combined with multifrontal methods the solver has near linear cost in examples due to very small iteration numbers that are essentially independent of problem size and number of subdomains it is to our knowledge only the second method with this property next to the moving pml sweeping method

area and hausdorff dimension of sierpinski carpet julia sets we prove the existence of rational maps whose julia sets are sierpinski carpets having positive area such rational maps can be constructed such that they either contain a cremer fixed point a siegel disk or are infinitely renormalizable we also construct some sierpinski carpet julia sets with zero area but with hausdorff dimension two moreover for any given number s in 1 2 we prove the existence of sierpinski carpet julia sets having hausdorff dimension exactly s

a criterion on the hyponormality of toeplitz operators with polynomial symbols via schur numbers in this paper we give a complete characterization on the hyponormality of the toeplitz operators t with trigonometric polynomial symbols g h g h h2 when g divides h this is accomplished by using a criterion on the contractivity of the inverse of a lower triangular finite toeplitz matrix via schur numbers of an induced analytic polynomial

optimal schwarz waveform relaxation for the one dimensional wave equation we introduce a nonoverlapping variant of the schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension we derive transmission conditions which lead to convergence of the algorithm in a number of iterations equal to the number of subdomains independently of the length of the time interval these optimal transmission conditions are in general nonlocal but we show that the nonlocality depends on the time interval under consideration and we introduce time windows to obtain optimal performance of the algorithm with local transmission conditions in the case of piecewise constant wave speed we show that convergence in two iterations can be achieved independently of the number of subdomains in that case the algorithm thus scales optimally with the number of subdomains provided the time windows are chosen appropriately for continuously varying coefficients we prove convergence of the algorithm with local transmission conditions using energy estimates we then introduce a finite volume discretization which permits computations on nonmatching grids and we prove convergence of the fully discrete schwarz waveform relaxation algorithm we finally illustrate our analysis with numerical experiments

one hundred years of complex dynamics holomorphic dynamics has been a key area of study for 100 years in this survey we focus on the dynamics of rational functions

analysis of the parallel schwarz method for growing chains of fixed sized subdomains part i in implicit solvation models the electrostatic contribution to the solvation energy can be estimated by solving a system of elliptic partial differential equations modeling the reaction potential the domain of definition of such elliptic equations is the union of the van der waals cavities corresponding to the atoms of the solute molecule therefore the computations can naturally be performed using schwarz methods where each atom of the molecule corresponds to a subdomain in contrast to classical schwarz theory it was observed numerically that the convergence of the schwarz method in this case does not depend on the number of subdomains even without coarse correction we prove this observation by analyzing the schwarz iteration matrices in fourier space and evaluating corresponding norms in a simplified setting in order to obtain our contraction results we had to choose a specific iteration formulation and we show that other formulations of the same algorithm can generate schwarz iteration matri

a gap between hyponormality and subnormality for block toeplitz operators this paper concerns a gap between hyponormality and subnormality for block toeplitz operators we show that there is no gap between 2 hyponormality and subnormality for a certain class of trigonometric block toeplitz operators e g its co analytic outer coefficient is invertible in addition we consider the extremal cases for the hyponormality of trigonometric block toeplitz operators in this case hyponormality and normality coincide

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

the sub cauchy stokes problem solvability issues and lagrange multiplier methods with artificial boundary conditions in contrast to the conventional cauchy stokes problems in which the velocity and the stress force data are given on the accessible boundary in the present paper we reduce the accessible boundary data information and we consider a problem which deals only with shear stress data we refer to this problem as a sub cauchy stokes problem this problem is ill posed because of severe instability and even uniqueness is unknown we first address the uniqueness issues associated with this problem resorting to the domain decomposition techniques together with the duplication process of vogelius 1 we propose new lagrange multiplier methods to solve the sub cauchy stokes problem these methods consist in recasting the problem in terms of interfacial equations by equalizing two solutions of the sub cauchy stokes problem using matching conditions defined on the inaccessible boundary the matching is based on second order conditions and depends on the equations used to match the values of the unknowns on the inaccessible boundary the underlying interfacial problems are then solved by iterative procedures in which coefficients can be optimized to improve convergence rates a complete analysis of the methods is presented and various numerical results illustrate the effectiveness and the performance of the proposed approaches

on a parallel robin type nonoverlapping domain decomposition method in recent years a nonoverlapping robin type domain decomposition method ddm for the finite element discretization systems of the second order elliptic equations which is based on using robin type boundary conditions as information transmission conditions on the subdomain interfaces has been developed and analyzed since it was first proposed by p l lions in on the schwarz alternating method iii a variant for nonoverlapping subdomains in proceedings of the 3rd international symposium on domain decomposition methods for partial differential equations siam philadelphia pa 1990 pp 202 223 however the convergence rate of this ddm with many subdomains remains open when the lower term of equations vanishes this open problem will be considered in this paper the convergence rate is almost 1 o h 1 2 h 1 2 in certain cases for example the case of a small number of subdomains where h is the mesh size and h is the size of subdomain in order to get the desirous convergence results two ma

the lyapunov exponent of holomorphic maps we prove that for any polynomial map with a single critical point its lower lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle similar statement holds for the exponential maps and some other complex dynamical systems we prove further that for the unicritical polynomials with positive area julia sets almost every point of the julia set has zero lyapunov exponent part of this statement generalizes as follows every point with positive upper lyapunov exponent in the julia set of an arbitrary polynomial is not a lebegue density point

a transcendental julia set of dimension 1 we construct a non polynomial entire function whose julia set has finite 1 dimensional spherical measure and hence hausdorff dimension 1 in 1975 baker proved the dimension of such a julia set must be at least 1 but whether this minimum could be attained has remained open until now our example also has packing dimension 1 and is the first transcendental julia set known to have packing dimension strictly less than 2 it is also the first example with a multiply connected wandering domain where the dynamics can be completely described

on the julia set of a typical quadratic polynomial with a siegel disk let 0 1 be an irrational number with continued fraction expan sion a1 a 2 a 3 and consider the quadratic polynomial p z 7 e 2i z z 2 by performing a trans quasiconformal surgery on an associated blaschke product model we prove that if log an o p n a sn 1 then the julia set of p is locally connected and has lebesgue measure zero in particular it follows that for almost every 0 1 the quadratic p has a siegel disk whose boundary is a jordan curve passing through the critical point of p b y standard renormalization theory these results generalize to the quadratics which have siegel disks of higher periods

optimized schwarz methods for maxwell s equations over the last two decades classical schwarz methods have been extended to systems of hyperbolic partial differential equations using characteristic transmission conditions and it has been observed that the classical schwarz method can be convergent even without overlap in certain cases this is in strong contrast to the behavior of classical schwarz methods applied to elliptic problems for which overlap is essential for convergence more recently optimized schwarz methods have been developed for elliptic partial differential equations these methods use more effective transmission conditions between subdomains than the classical dirichlet conditions and optimized schwarz methods can be used both with and without overlap for elliptic problems we show here why the classical schwarz method applied to both the time harmonic and time discretized maxwell s equations converges without overlap the method has the same convergence factor as a simple optimized schwarz method for a scalar elliptic equation based on this insight we develop an entire new hierarchy of optimized overlapping and nonoverlapping schwarz methods for maxwell s equations with greatly enhanced performance compared to the classical schwarz method we also derive for each algorithm asymptotic formulas for the optimized transmission conditions which can easily be used in implementations of the algorithms for problems with variable coefficients we illustrate our findings with numerical experiments

robin robin domain decomposition methods for the stokes darcy coupling in this paper we consider a coupled system made of the stokes and darcy equations and we propose some iteration by subdomain methods based on robin conditions on the interface we prove the convergence of these algorithms and for suitable finite element approximations we show that the rate of convergence is independent of the mesh size h special attention is paid to the optimization of the performance of the methods when both the kinematic viscosity nu of the fluid and the hydraulic conductivity tensor k of the porous medium are very small

normal singular cauchy integral operators with operator valued symbols abstract in this paper we characterize a class of normal isometric coisometric unitary hyponormal singular cauchy integral operators with operator valued or matrix valued symbols on vector valued or c n valued l 2 spaces

subnormal toeplitz operators and the kernels of their self commutators abstract in this note we give a connection between subnormal toeplitz operators and the kernels of their self commutators this is closely related to p r halmos s problem 5 is every subnormal toeplitz operator either normal or analytic our main theorem is as follows if l is such that and are of bounded type that is they are quotients of two analytic functions on the open unit disk and if the kernel of the self commutator of t is invariant for t then t is either normal or analytic

hyponormality of toeplitz operators with polynomial symbols a bounded linear operator a on a hilbert space h with inner product is said to be hyponormal if its selfcommutator a a a a aa induces a positive semidefinite quadratic form on h via 7 a a for h let h t denote the hardy space of the unit circle t d in the complex plane recall that given l t the toeplitz operator with symbol is the operator t on h t defined by t f p f where f h t and p denotes the projection that maps l t onto h t the hyponormality of toeplitz operators has been studied by c cowen 1 2 p fan 4 c gu 8 t ito and t wong 9 t nakazi and k takahashi 11 d yu 13 k zhu 14 r curto d farenick the second and the third named authors 3 5 6 10 and others an elegant theorem of c cowen 2 characterizes the hyponormality of a toeplitz operator t on h t by properties of the symbol l t k zhu 14 reformulated cowen s criterion and then showed that the hyponormality of t with polynomial symbols can be decided by a method based on the classical interpolation theorem of i schur 12 also farenick and the third named author 5 characterized the hyponormality of t in terms of the fourier coefficients of the trigonometric polynomial in the cases that the outer coefficients of have the same modulus but the case of arbitrary trigonometric polynomials though solved in principle by cowen s theorem or zhu s theorem is in practice very complicated on the other hand nakazi and takahashi 11 corollary 5 showed that if z n n m anz n is a trigonometric polynomial with m n and if for every zero of z such that 1 the number 1 is a zero of z in the open unit disk d of multiplicity greater than or equal to the multiplicity of then t is hyponormal but the converse is not true in general to see this consider the following trigonometric polynomial z z 2 z 2 z 1 z 15 z 13 then z 2 15z 2 19 15z 55 15 53 15z z using an argument of p fan 4 theorem 1 for every trigonometric polynomial of the form z 2 n 2 anz

one way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems non overlapping domain decomposition dd methods with complex first order robin type transmission conditions tcs provide an efficient iterative solution for maxwell s equation unfortunately the first order tcs do not effectively account for some eigenmodes of the system matrix which limits the scalability of the methods in this work we examine two tcs with a second order transverse derivative to improve the method s performance a detailed convergence analysis of the two tcs is presented we then investigate the use of the two second order tcs in non conformal and non overlapping one way dd methods numerical results illustrate the effectiveness of the proposed methods on some model problems and on several problems of practical interest

fine structure of connectedness loci we study distributions of the harmonic and lebesgue measures on the boundary of the connectedness locus mathcal m d for uni critical polynomials z d c and dynamical properties which are typical with respect to the harmonic measure one result is a conformal similarity between mathcal m d and the corresponding julia set for almost every point of mathcal m d with respect to the harmonic measure it is also shown that the harmonic measure is supported on lebesgue density points of the complement of mathcal m d which are not accessible from outside within john angles and at which the boundary of mathcal m d spirals infinitely often in both directions in the case of quadratic polynomials we prove that every parameter c from the boundary of the mandelbrot set mathcal m that is recurrent under iterates of z 2 c but not infinitely tunable is a lebesgue weak density point of the complement of mathcal m a direct consequence of our theory is the yoccoz local connectivity theorem a new technical ingredient in the paper is the method of amplification which is based on promoting the small scale geometry of the parameter space at a typical point in the sense of harmonic measure to uniform geometry of the corresponding julia set the pointwise conformality of the similarity map is derived through inducing techniques and a classical twb theory

hyponormality of toeplitz operators with rational symbols in this article we introduce a notion of division for rational functions and then give a criterion for hyponormality of open image in new window f g are rational functions in the cases where g divides f furthermore we show that we may assume without loss of generality that g divides f when we consider the hyponormality of open image in new window

which subnormal toeplitz operators are either normal or analytic abstract we study subnormal toeplitz operators on the vector valued hardy space of the unit circle along with an appropriate reformulation of p r halmos s problem 5 which subnormal block toeplitz operators are either normal or analytic we extend and prove abrahamse s theorem to the case of matrix valued symbols that is we show that every subnormal block toeplitz operator with bounded type symbol i e a quotient of two bounded analytic functions whose analytic and co analytic parts have the left coprime factorization is normal or analytic we also prove that the left coprime factorization condition is essential finally we examine a well known conjecture of whether every subnormal toeplitz operator with finite rank self commutator is normal or analytic

absorbing boundary conditions for the wave equation and parallel computing absorbing boundary conditions have been developed for truncating unbounded computational domains to bounded ones the authors investigate the use of absorbing boundary conditions in domain decomposition algorithms for the wave equation moreover they optimize the absorbing boundary conditions in order to obtain a good performance of the schwarz waveform relaxation algorithm the analysis given is confirmed by numerical experiments

optimized schwarz methods for model problems with continuously variable coefficients optimized schwarz methods perform better than classical schwarz methods because they use more effective transmission conditions between subdomains these transmission conditions are determined by optimizing the convergence factor which is obtained by fourier analysis for simple two subdomain model problems such optimizations have been performed for many different types of partial differential equations but almost exclusively based on the assumption of constant coefficients because only then fourier analysis can be applied we use in this paper the technique of separation of variables to study optimized schwarz methods for a model problem with a continuously variable reaction term and a similar analysis could be performed as well for many other problems with variable coefficients we obtain several new interesting results first we show that the technique of separation of variables can successfully decouple the spatial variables and give the convergence factor of subdomain iterations as a function of

quadratic julia sets with positive area we recently proved the existence of quadratic polynomials having a julia set with positive lebesgue measure we present the ideas of the proof and the techniques involved

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

non overlapping schwarz methods with optimized transmission conditions for the helmholtz equation abstract the development of efficient transmission conditions for the schwarz method without overlap is an important area of current research the techniques developed in recent years mainly consist of algebraic approximations of the discrete transparent operator or optimized approximations of the continuous transparent operator these optimized approximations are currently based on an analytical approach involving a minimization problem over all the available frequencies and using the l norm in this paper new numerical optimization techniques are investigated these original approaches allow selection of the frequency range for the minimization problem which is based on the l p norm new numerical experiments performed on a model problem demonstrate the performance and the robustness of the proposed optimization techniques in the particular cases p 1 and p applications to industrial automotive problems confirm the efficiency of these techniques when using arbitrary mesh partitions

optimized schwarz methods for the diffusion reaction problem with cylindrical interfaces in this work we consider the optimized schwarz method for the three dimensional 3d diffusion reaction problem in particular we treat the case of cylindrical interfaces between the subdomains and we provide a convergence analysis of the schwarz method both in the case of dirichlet interface conditions and in that of general transmission conditions this allows us to recover for the latter case optimal symbols for the interface conditions which are supposed to work well for geometries which feature cylindrical interfaces moreover starting from these optimal symbols we propose effective and easily computable constant interface parameters derived both from taylor expansions and from an optimization procedure we finally present several 3d numerical results aiming at validating the theoretical findings

condition number estimates for the nonoverlapping optimized schwarz method and the 2 lagrange multiplier method for general domains and cross points the optimized schwarz method and the closely related 2 lagrange multiplier method are domain decomposition methods which can be used to parallelize the solution of partial differential equations although these methods are known to work well in special cases e g when the domain is a square and the two subdomains are rectangles the problem has never been systematically stated nor analyzed for general domains with general subdomains the problem of cross points when three or more subdomains meet at a single vertex has been particularly vexing we introduce a 2 lagrange multiplier method for domain decompositions with cross points we estimate the condition number of the iteration and provide an optimized robin parameter for general domains we hope that this new systematic theory will allow broader utilization of optimized schwarz and 2 lagrange multiplier preconditioners

optimized schwarz methods for circular domain decompositions with overlap optimized schwarz methods are based on transmission conditions between subdo mains which are optimized for the problem class that is being solved such optimizations have been performed for many different types of partial differential equations but were almost exclusively based on the assumption of straight interfaces we study in this paper the influence of curvature on the optimization and we obtain four interesting new results first we show that the curvature does indeed enter the optimized parameters and the contraction factor estimates second we develop an asymptotically accurate approximation technique based on turan type inequalities in our case to solve the much harder optimization problem on the curved interface and this approximation tech nique will also be applicable to currently too complex best approximation problems in the area of optimized schwarz methods third we show that one can obtain transmission conditions from a simple circular model decomposition which have also been found using microlocal analysis but that these are not the best choices for the performance of the optimized schwarz method finally we find that in the case of curved interfaces optimized schwarz methods are not necessarily convergent for all admissible parameters our optimization leads however to parameter choices that give the same good performance for a circular decomposition as for a straight interface decomposition we illustrate our analysis with numerical experiments

optimized schwarz methods optimized schwarz methods are a new class of schwarz methods with greatly enhanced convergence properties they converge uniformly faster than classical schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical schwarz methods if the overlap is of the order of the mesh parameter which is often the case in practical applications they achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem we analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new finite element tearing and interconnect feti variants

on the geometric convergence of optimized schwarz methods with applications to elliptic problems the schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain one subdivides into smaller more manageable subdomains and solves the differential equation in these subdomains using appropriate boundary conditions optimized schwarz methods use robin conditions on the artificial interfaces for information exchange at each iteration and for which one can optimize the robin parameters while the convergence theory of classical schwarz methods with dirichlet conditions on the artificial interface is well understood the overlapping optimized schwarz methods still lack a complete theory in this paper an abstract hilbert space version of the optimized schwarz method osm is presented together with an analysis of conditions for its geometric convergence it is also shown that if the overlap is relatively uniform these convergence conditions are met for optimized schwarz methods for two dimensional elliptic problems for any positive robin parameter in the discrete setting we obtain that the convergence factor h varies like a polylogarithm of h numerical experiments show that the methods work well and that the convergence factor does not appear to depend on h

optimized schwarz methods for the time harmonic maxwell equations with damping in a previous paper two of the authors have proposed and analyzed an entire hierarchy of optimized schwarz methods for maxwell s equations in both the time harmonic and the time domain case the optimization process has been performed in a particular situation where the electric conductivity was neglected here we take into account this physical parameter which leads to a fundamentally different analysis and a new class of algorithms for this more general case from the mathematical point of view the approach is different since the algorithm does not encounter the pathological situations of the zero conductivity case and thus the optimization problems are different we analyze one of the algorithms in this class in detail and provide asymptotic results for the remaining ones we illustrate our analysis with numerical results

hyponormality and subnormality of block toeplitz operators abstract in this paper we are concerned with hyponormality and subnormality of block toeplitz operators acting on the vector valued hardy space h c n 2 of the unit circle first we establish a tractable and explicit criterion on the hyponormality of block toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator second we consider the gap between hyponormality and subnormality for block toeplitz operators this is closely related to halmos s problem 5 is every subnormal toeplitz operator either normal or analytic we show that if is a matrix valued rational function whose co analytic part has a coprime factorization then every hyponormal toeplitz operator t whose square is also hyponormal must be either normal or analytic third using the subnormal theory of block toeplitz operators we give an answer to the following toeplitz completion problem find the unspecified toeplitz entries of the partial block toeplitz matrix a u u so that a becomes subnormal where u is the unilateral shift on h 2

hyponormality of trigonometric toeplitz operators in this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric toeplitz operators i e toeplitz operators t with trigonometric polynomial symbols our criterion involves the zeros of an analytic polynomial f induced by the fourier coefficients of moreover the rank of the selfcommutator of t is computed from the number of zeros of f in the open unit disk d and in c d counting multiplicity

block toeplitz operators with rational symbols ii in this paper we derive a formula for the rank of the self commutator of hyponormal block toeplitz operators t with matrix valued rational symbols in via the classical hermite fej r interpolation problem

upper bound for the size of quadratic siegel disks if is an irrational number we let p n q n n 0 be the approximants given by its continued fraction expansion the bruno series b is defined as b alpha sum n geq 0 frac log q n 1 q n the quadratic polynomial p z e 2i z z 2 has an indifferent fixed point at the origin if p is linearizable we let r be the conformal radius of the siegel disk and we set r 0 otherwise yoccoz proved that if b then r 0 and p is not linearizable in this article we present a different proof and we show that there exists a constant c such that for all irrational number with b we have b alpha log r alpha c together with former results of yoccoz see y this proves the conjectured boundedness of b logr

a proof of the marmi moussa yoccoz conjecture for rotation numbers of high type marmi moussa and yoccoz conjectured that some error function upsilon related to the approximation of the size of siegel disks by some arithmetic function of the rotation number theta is a holder continuous function of theta with exponent 1 2 using the renormalization invariant class of inou and shishikura we prove this conjecture for the restriction of upsilon to a class of high type numbers

new nonoverlapping domain decomposition methods for the harmonic maxwell system we study a nonoverlapping domain decomposition method for the harmonic maxwell equations with a new kind of interface condition using fourier analysis we derive suitable families of transmission conditions in r3 that involve second order tangential differential operators and that guarantee convergence for both propagative and evanescent modes such families depend upon parameters that are chosen to optimize the convergence rate of the corresponding iterative algorithm we then propose iterative solvers for the maxwell equations based on a domain decomposition procedure where such conditions are enforced on the interface some numerical results for a two domain decomposition show the effectiveness of the optimized interface conditions

kernels of hankel operators and hyponormality of toeplitz operators we give a formula for mathrm ker h overline theta 1 ast h overline theta 2 and describe when mathrm ker h overline theta 1 ast h overline theta 2 mathrm ker h overline theta 2 we explore the hyponormality of toeplitz operators whose symbols are of circulant type and some more general types in addition we discuss formulas for and estimates of the rank of the self commutator of a hyponormal toeplitz operator

optimized schwarz methods with ventcell transmission conditions for model problems with continuously variable coefficients abstract the ventcell transmission conditions can enhance greatly the convergence of subdomain iterations using optimization based parameters however such optimization can now only be performed for model problems with constant coefficients and thus forms a gap between the theoretical analysis and the applications where the model problems are seldom to be constant coefficients in this paper we close this gap by a rigorous analysis on the optimized schwarz methods with ventcell transmission conditions for a model problem where the reaction term varies parallel to the interface continuously we obtain the asymptotically optimal transmission parameters by optimizing a convergence factor that is obtained using the technique of separation of variables other than the often used fourier transform and then the corresponding convergence factor estimate follows we at last validate the theoretical findings by numerical experiments

finite element formulation in flat coordinate spaces to solve elliptic problems on general closed riemannian manifolds we apply finite element methods to elliptic problems on closed riemannian manifolds the elliptic equations on manifolds are reduced to coupled equations on euclidean spaces by using coordinate charts the advantage of this strategy is to avoid global triangulations on curved manifolds in the finite element method the resulting finite element problems can be solved by direct methods or by domain decomposition iterations we present a convergence theory and some computational results our methods are illustrated by a 3 dimensional sphere problem in 4 dimensional euclidean space we also give a numerical comparison between the surface finite element method and our methods

convergence estimates for an higher order optimized schwarz method for domains with an arbitrary interface optimized schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations optimized schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence in the literature the analysis of optimized schwarz methods relies on fourier analysis and so the domains are restricted to be regular rectangular in this paper we express the interface operator of an optimized schwarz method in terms of poincare steklov operators this enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1 o h 1 4 on a class of general domains where h is the discretization parameter this is the predicted rate for a second order optimized schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations

a two grid method of a mixed stokes darcy model for coupling fluid flow with porous media flow we study numerical methods for solving a coupled stokes darcy problem in porous media flow applications a two grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions error estimates are derived for the proposed method both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two grid approach for solving multimodeling problems potential extensions and future directions are discussed

locally conservative coupling of stokes and darcy flows a locally conservative numerical method for solving the coupled stokes and darcy flows problem is formulated and analyzed the approach employs the mixed finite element method for the darcy region and the discontinuous galerkin method for the stokes region a discrete inf sup condition and optimal error estimates are derived

timely communicaton an analysis for a nonoverlapping domain decomposition iterative procedure an iterative procedure for nonoverlapping domain decomposition methods ddm is developed and analyzed which can be parallelly implemented with local communications it uses a robin boundary condition as its transmission condition on the interfaces the update of robin data of its transmission condition is a very simple relaxation between the preceding robin data and function value it need not find any derivatives at each iterative step this method can be easily applied to the discretization problems moreover it is shown that it is indeed a variant of the lions nonoverlapping ddm for the continuous problems also it can be regarded as a generalization of tang s generalized schwarz alternating method gsam in the nonoverlapping ddm

a source transfer domain decomposition method for helmholtz equations in unbounded domain we propose and study a domain decomposition method for solving the truncated perfectly matched layer pml approximation in bounded domain of helmholtz scattering problems the method is based on the decomposition of the domain into nonoverlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a pml method defined locally outside the last two layers the convergence of the method is proved for the case of constant wave number based on the analysis of the fundamental solution of the pml equation the method can be used as an efficient preconditioner in the preconditioned gmres method for solving discrete helmholtz equations with constant and heterogeneous wave numbers numerical examples are included