almost exponential decay of periodic viscous surface waves without surface tension we consider a viscous fluid of finite depth below the air occupying a three dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary the fluid dynamics are governed by the gravity driven incompressible navier stokes equations and the effect of surface tension is neglected on the free surface the long time behavior of solutions near equilibrium has been an intriguing question since the work of beale commun pure appl math 34 3 359 392 1981 this paper is the third in a series of three guo in local well posedness of the viscous surface wave problem without surface tension anal pde 2012 to appear in decay of viscous surface waves without surface tension in horizontally infinite domains preprint 2011 that answers this question here we consider the case in which the free interface is horizontally periodic we prove that the problem is globally well posed and that solutions decay to equilibrium at an almost exponential rate in particular the free interface decays to a flat surface our framework contains several novel techniques which include 1 a priori estimates that utilize a geometric reformulation of the equations 2 a two tier energy method that couples the boundedness of high order energy to the decay of low order energy the latter of which is necessary to balance out the growth of the highest derivatives of the free interface 3 a localization procedure that is compatible with the energy method and allows for curved lower surface geometry our decay estimates lead to the construction of global in time solutions to the surface wave problem

a finite element spectral method for approximating the time harmonic maxwell system in r 3 we describe and analyze a method for computing an approximation to the time harmonic electromagnetic field scattered by a bounded inhomogeneity the method is to couple a finite element scheme on a bounded domain with a series solution outside the bounded domain the main result of the paper is to show that the proposed numerical scheme possesses a unique solution with quasi optimal approximation properties we do this by verifying the conditions of the babuska brezzi theory for saddle point problems in analyzing the continuous problem we also provide a new variational proof of the existence of solutions of the continuous maxwell problem

inverse medium scattering problems for electromagnetic waves consider a time harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in mathbb r 3 in this paper existence and uniqueness of the variational problem for forward scattering are established an energy estimate for the scattered field with a uniform bound with respect to the wavenumber is obtained in the case of low frequency on which the born approximation is based a continuation method for the inverse medium scattering problem which reconstructs the scatterer of an inhomogeneous medium from boundary measurements of the scattered wave is developed the algorithm requires multifrequency scattering data using an initial guess from the born approximation each update is obtained via recursive linearization on the wavenumber k by solving one forward problem and one adjoint problem of maxwell s equations

corrigendum a finite element spectral method for approximating the time harmonic maxwell system in r 3 in our paper siam j appl math 55 1995 pp 1234 1344 theorem 4 2 b is incorrect this note shows how to avoid the use of theorem 4 2 b in the remainder of the paper we emphasize that the main results of the paper theorems 4 6 and 5 3 are correct but that the proofs must be modified as shown in this note

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

on stabilizing effect of elasticity in the rayleigh taylor problem of stratified viscoelastic fluids abstract we investigate the stabilizing effect of elasticity in the rayleigh taylor rt problem of stratified immiscible viscoelastic fluids separated by a free interface and in the presence of a uniform gravitational field in a horizontally periodic domain where the velocities of the fluids are non slip on both upper and lower fixed flat boundaries while the internal surface tension is omitted we establish a discriminant c r for the stability of the stratified viscoelastic rt problem more precisely if c r 1 then the stratified viscoelastic rt equilibrium state is exponentially stable this means that a sufficiently large elasticity coefficient has stabilizing effect so that it can inhibit viscoelastic rt instability on the other hand if c r 1 then we show that the rt equilibrium state is linearly unstable in the hadamard sense moreover for the case of a nonhomogeneous incompressible viscoelastic fluid the condition c r 1 will lead to the nonlinear instability of the rt equilibrium state and this shows that the rt instability still occurs in viscoelastic fluids when the elasticity coefficient is small

global existence and decay of smooth solution for the 2 d mhd equations without magnetic diffusion abstract we prove the global existence and the decay estimates of small smooth solution for the 2 d mhd equations without magnetic diffusion this confirms the numerical observation that the energy of the mhd equations is dissipated at a rate independent of the ohmic resistivity

an extension of the positive definite and skew hermitian splitting method for preconditioning of generalized saddle point problems abstract we extend the positive definite and skew hermitian splitting pss iterative method to epss for extended pss for solving non hermitian positive definite systems of linear equations in this kind of extension the shift matrix is replaced by a hermitian positive definite matrix it is proved that the method is unconditionally convergent then the epss method with a 2 2 block diagonal matrix as the shift matrix is applied to generalized saddle point problems and the convergence of the method is verified naturally the induced preconditioner can be applied to generalized saddle point problems in some special cases this preconditioner coincides with some existing preconditioners a special case of the epss preconditioner say sepss is used to accelerate the convergence of gmres effectiveness of this preconditioner is investigated by some numerical experiments

marching schemes for inverse acoustic scattering problems for the numerical solution of inverse helmholtz problems the boundary value problem for a helmholtz equation with spatially variable wave number has to be solved repeatedly for large wave numbers this is a challenge in the paper we reformulate the inverse problem as an initial value problem and describe a marching scheme for the numerical computation that needs only n2 log n operations on an n n grid we derive stability and error estimates for the marching scheme we show that the marching solution is close to the low pass filtered true solution we present numerical examples that demonstrate the efficacy of the marching scheme

exponential decay of benard convection problem with surface tension abstract we consider the dynamics of an boussinesq approximation benard convection fluid evolving in a three dimensional domain bounded below by a fixed flatten boundary and above by a free moving surface the domain is horizontally periodic and the effect of the surface tension is on the free surface by developing a priori estimates for the model we prove the exponential decay of solutions in the framework of high regularity

a relaxed deteriorated pss preconditioner for nonsymmetric saddle point problems from the steady navier stokes equation for nonsymmetric saddle point problems arising from the steady navier stokes equations pan ng and bai presented a deteriorated positive definite and skew hermitian splitting dpss preconditioner pan et al 2006 to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the unconditional convergence property of the dpss iteration method is proved and a relaxed dpss rdpss preconditioner is proposed the rdpss preconditioner is much closer to the coefficient matrix than the dpss preconditioner in certain norm which straightforwardly results in an rdpss iteration method the convergence conditions of the rdpss iteration are analyzed and the optimal parameter which minimizes the spectral radius of the rdpss iteration matrix is derived using the rdpss preconditioner to accelerate some krylov subspace methods like gmres is also studied the eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the rdpss preconditioner

on inexact preconditioners for nonsymmetric matrices inexact versions of the block triangular preconditioners for nonsymmetric matrices of block two by two structures in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 and i c f ipsen siam j sci comput 23 2001 pp 1050 1051 are presented and the two preconditioners for symmetric block two by two matrices in c durazzi and v ruggiero numer linear algebra appl 10 2003 pp 673 688 are extended to general nonsymmetric matrices moreover we precisely describe the spectral properties of the preconditioned matrices and the finite step termination properties of the preconditioned krylov subspace iteration methods with an optimal or galerkin property with respect to these preconditioners several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners

numerical resolution of an electromagnetic inverse medium problem at fixed frequency the aim of this paper is to solve numerically the inverse problem of determining the complex refractive index of an electromagnetic medium from partial boundary field measurements at a fixed frequency the governing equations are the time harmonic maxwell equations formulated in electric field in a two dimensional bounded domain we express the inverse problem as the minimization of a cost function representing the difference between the measured and predicted fields our numerical reconstruction algorithm combines the bfgs method and an iterative process called the adaptive eigenspace inversion the unknown complex coefficient is expanded in terms of eigenfunctions of an elliptic operator both the eigenspace and the mesh are iteratively adapted during the minimization procedure numerical experiments illustrate the performance of the reconstruction for various configurations

a relaxed dimensional factorization preconditioner for the incompressible navier stokes equations in this paper we introduce a relaxed dimensional factorization rdf preconditioner for saddle point problems properties of the preconditioned matrix are analyzed and compared with those of the closely related dimensional splitting ds preconditioner recently introduced by benzi and guo 7 numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the rdf preconditioner with respect to both mesh size and viscosity

numerical solution of inverse scattering problems with multi experimental limited aperture data in this paper a regularized homotopy continuation method is presented for numerical solution of nonlinear ill posedness inverse problems two major difficulties for solving these inverse problems by an optimization method are the ill posedness and the presence of many local minima classic iterative methods such as gauss newton or levenberg marquardt algorithms offer fast local convergence but might not be able to compute the global minimum based on a natural concept of multi experimental data a regularized homotopy continuation method is constructed to compute the global minimum as the experimental index t increases continuously the global minimizer can be computed continuously by using local optimization methods by discretizing the continuous homotopy method various recursive linearization algorithms are developed these algorithms are applied to numerical solution of an inverse medium scattering problem which reconstructs the refractive index of an inhomogeneous medium from limited aperture measurements of the far field pattern of the scattered fields it is assumed that the data is measured at multiple frequencies the inverse scattering problem with limited aperture is challenging since without full aperture measurements the ill posedness and nonlinearity of the inverse problem become more severe an efficient regularized iterative linearization method recursive linearization with respect to the wave number is developed for solving the inverse problem the convergence of the iterative method is examined by numerical examples

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

on the free boundary problem to the two viscous immiscible fluids abstract in this paper we prove the local solvability of the free boundary problem describing the motion of two layers of immiscible heavy viscous incompressible fluid lying above an infinite rigid bottom and with surface tension on the interfaces and global solvability near the equilibrium state

zero surface tension limit of the free surface incompressible euler equations with damping abstract we consider the free boundary problem for a layer of an incompressible inviscid fluid in a uniform gravitational field lying above a rigid bottom and below the atmosphere in a horizontally periodic setting we establish the global in time zero surface tension limit of the problem with damping for the small initial data

a modified block preconditioner for complex nonsymmetric indefinite linear systems abstract we propose a modified block splitting preconditioner for a class of complex nonsymmetric indefinite linear systems by adopting two iteration parameters and a relaxing technique the new preconditioner is much closer to the original coefficient matrix theoretical analysis proves that the preconditioned matrix has an eigenvalue 1 with algebraic multiplicity at least n a theorem concerning the dimension of the krylov subspace for the preconditioned matrix is also obtained finally some numerical experiments are presented to illustrate the effectiveness of the preconditioner presented

recent developments in inverse acoustic scattering theory we survey some of the highlights of inverse scattering theory as it has developed over the last 15 years with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects

a note on the positive stable block triangular preconditioner for generalized saddle point problems abstract in this paper we give a sharp eigenvalue bound for the positive stable block triangular preconditioned matrix presented in a recent paper by cao z h cao positive stable block triangular preconditioners for symmetric saddle point problems appl numer math 57 2007 899 910 the intervals containing these eigenvalues of the preconditioned matrix remove the origin which is benefit for further study numerical experiments of a model stokes problem are presented to show the estimate and the effectiveness of the positive stable block triangular preconditioners

the generalized polarization tensors for resolved imaging part ii shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements this paper deals with the problem of reconstructing the electromagnetic parameters and the shape of a target from multi static response matrix measurements at a single frequency the target is of characteristic size less than the operating wavelength using long wavelength asymptotic expansions of the measurements of high order we show how the electromagnetic parameters and the equivalent ellipse of the target can be reconstructed the asymptotic expansions of the measurements are written in terms of the new concept of frequency dependent polarization tensors moreover we extend the optimization approach proposed in part i 9 to image geometric details of an electromagnetic target that are finer than the equivalent ellipse the equivalent ellipse still provides a good initial guess for the optimization procedure however compared to the conductivity case in 9 the cost functional measures the discrepancy between the computed and measured high order frequency dependent polarization tensors rather than between the generalized polarization tensors the main reason for such a modification of the cost functional is the fact that the measured frequency dependent polarization tensors can be easily obtained from multistatic measurements by solving a linear system while the derivation of the generalized polarization tensors from measurements requires more delicate treatment the proposed methods are numerically implemented to demonstrate their validity and efficiency mathematics subject classification msc2000 35r30 35b30

global regularity of 2d density patches for inhomogeneous navier stokes this paper is about lions open problem on density patches lions in mathematical topics in fluid mechanics vol 1 volume 3 of oxford lecture series in mathematics and its applications clarendon press oxford university press new york 1996 whether or not inhomogeneous incompressible navier stokes equations preserve the initial regularity of the free boundary given by density patches using classical sobolev spaces for the velocity we first establish the propagation of c 1 gamma regularity with 0 gamma 1 in the case of positive density furthermore we go beyond this to show the persistence of a geometrical quantity such as the curvature in addition we obtain a proof for c 2 gamma regularity

a finite element method for approximating electromagnetic scattering from a conducting object we provide an error analysis of a fully discrete finite element fourier series method for approximating maxwell s equations the problem is to approximate the electromagnetic field scattered by a bounded inhomogeneous and anisotropic body the method is to truncate the domain of the calculation using a series solution of the field away from this domain we first prove a decomposition for the poincare steklov operator on this boundary into an isomorphism and a compact perturbation this is proved using a novel argument in which the scattering problem is viewed as a perturbation of the free space problem using this decomposition and edge elements to discretize the interior problem we prove an optimal error estimate for the overall problem

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

a convergence analysis of the landweber iteration for nonlinear ill posed problems in this paper we prove that the landweber iteration is a stable method for solving nonlinear ill posed problems for perturbed data with noise level delta we propose a stopping rule that yields the convergence rate o delta 1 2 under appropriate conditions we illustrate these conditions for a few examples

a relaxed positive definite and skew hermitian splitting preconditioner for saddle point problems for saddle point linear system whose coefficient matrix is non hermitian of positive definite hermitian part pan et al 2006 proposed a positive definite and skew hermitian splitting pss preconditioner in this paper based on the pss preconditioner we present a relaxed positive definite and skew hermitian splitting rpss preconditioner we analyze the spectral properties of the new preconditioned matrix and conclude that the rpss preconditioner has much better properties than pss numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the new preconditioner

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

reconstruction of two dimensional permittivity distribution using the distorted born iterative method the distorted born iterative method dbim is used to solve two dimensional inverse scattering problems thereby providing another general method to solve the two dimensional imaging problem when the born and the rytov approximations break down numerical simulations are performed using the dbim and the method proposed previously by the authors int j imaging syst technol vol 1 no 1 p 100 8 1989 called the born iterative method bim for several cases in which the conditions for the first order born approximation are not satisfied the results show that each method has its advantages the dbim shows faster convergence rate compared to the bim while the bim is more robust to noise contamination compared to the dbim

instability solutions for the rayleigh taylor problem of non homogeneous viscoelastic fluids in bounded domains abstract it is well known that the rayleigh taylor rt problem of an inhomogeneous viscoelastic fluid defined on a bounded domain is unstable if the elasticity coefficient is less than some threshold c in this paper we rigorously prove the existence of a unique unstable strong solution in the sense of l 1 norm for the rt problem in lagrangian coordinates based on a bootstrap instability method when c applying an inverse transformation of lagrangian coordinates to the obtained unstable solution we can further get a unique unstable solution for the rt problem of inhomogeneous viscoelastic fluids in eulerian coordinates

global well posedness of the free surface incompressible euler equations with damping abstract we consider a layer of an incompressible inviscid fluid bounded below by a fixed solid boundary and above by a free moving boundary in a horizontally periodic setting the fluid dynamics is governed by the gravity driven incompressible euler equations with damping and the effect of surface tension is neglected on the free surface we prove that the problem is globally well posed for the small initial data and that solutions decay to the equilibrium at an almost exponential rate

application of the fem with adaptivity for electromagnetic inverse medium scattering problems abstract the paper presents a study of solution of inverse medium scattering problems for 3d time harmonic electromagnetics the solution is based on minimization of the misfit function between the observed scattered fields and trial solutions of direct problems for the sought approximate distribution of electric permittivity the sources of illuminating waves and the observation points of the scattered fields are located in the nearfield the scatterer the simulations of the direct problems are performed with the finite element method application of adaptive mesh refinement to improve accuracy of solutions is illustrated tests with real and complex valued distributions of electric permittivity and with complete and incomplete measurement data are presented

a relaxed upper and lower triangular splitting preconditioner for the linearized navier stokes equation abstract based on the block triangular splitting of the original coefficient matrix a relaxed upper and lower triangular splitting rult preconditioner for the linearized incompressible navier stokes equations is considered the convergence analysis of the rult iteration method is presented the spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed respectively then the quasi optimal parameters of the rult preconditioner are derived finally some numerical results are carried out to show the effectiveness of the rult preconditioner and verify that the rult preconditioner outperforms the mds grs mrs and rbts preconditioners for the gmres method for solving the linearized incompressible navier stokes equation in terms of the number of iterations and computation times

global wellposedness of the 3 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 3 d infinite depth full water wave problem in the setting that the interface tends to the horizontal plane the velocity and acceleration on the interface tend to zero at spatial infinity we show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders for any initial interface that is sufficiently small in its steepness and velocity we show that there exists a unique smooth solution of the full water wave problem for all time and the solution decays at the rate 1 t

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

almost global wellposedness of the 2 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 2 d infinite depth full water wave equation it is known that this equation has a solution for a time period 0 t e for initial data of the form e where t depends only on in this paper we show that for such data there exists a unique solution for a time period 0 e t e this is achieved by better understandings of the nature of the nonlinearity of the full water wave equation

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

well posedness in sobolev spaces of the full water wave problem in 2 d we consider the motion of the interface of 2 d irrotational incompressible inviscid water wave with air above water and surface tension zero we show that the interface is always not accelerating into the water region normal to itself as rapidly as the normal acceleration of gravity as long as the interface is nonself intersect we therefore obtain the existence and uniqueness of solutions of the full water wave problem locally in time for any initial interface which is nonself intersect including the case that the interface is of multiple heights

fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem we propose a method for computing the scattering of a time harmonic electromagnetic wave in a medium with a locally perturbed refractive index we prove that it converges of arbitrarily high order depending on the smoothness of the refractive index and that the time complexity is o n 3log n as an application we solve the corresponding inverse problem using a preconditioned newton method and discuss how to compute the frechet derivative of the solution operator and its adjoint by the proposed method the performance of both the forward and the inverse solver is illustrated in a number of numerical experiments

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

well posedness of the free surface incompressible euler equations with or without surface tension we provide a new method for treating free boundary problems in perfect fluids and prove local in time well posedness in sobolev spaces for the free surface incompressible 3d euler equations with or without surface tension for arbitrary initial data and without any irrotationality assumption on the fluid this is a free boundary problem for the motion of an incompressible perfect liquid in vacuum wherein the motion of the fluid interacts with the motion of the free surface at highest order

numerical solution of an inverse medium scattering problem for maxwell s equations at fixed frequency consider a time harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in r 3 in this paper well posedness of the variational problem for the direct scattering is examined an energy estimate for the scattered field is obtained on which the born approximation is based a regularized recursive linearization method for the inverse medium scattering which reconstructs the scatterer of an inhomogeneous medium from the boundary measurements of the scattered field is developed the algorithm requires only single frequency data using an initial guess from the born approximation each update is obtained via continuation on the spatial frequency of a two parameter family of plane waves by solving one direct problem and one adjoint problem of the maxwell equation

on the free boundary problem of three dimensional incompressible euler equations in this paper we consider in three dimensions the motion of a general inviscid incompressible fluid with a free interface that separates the fluid region from the vacuum we assume that the fluid region is below the vacuum and that there is no surface tension on the free surface then we prove the local well posedness of the free boundary problem in sobolev space provided that there is no self intersection point on the initial surface and under the stability assumption that with being restricted to the initial surface 2007 wiley periodicals inc

new choices of preconditioning matrices for generalized inexact parameterized iterative methods for large sparse saddle point problems chen and jiang recently studied a class of generalized inexact parameterized iterative methods see f chen y l jiang a generalization of the inexact parameterized uzawa methods for saddle point problems appl math comput 206 2008 765 771 in this paper the methods are modified and some choices of preconditioning matrices are given these preconditioning matrices have advantages in solving large sparse linear system numerical experiments of a model stokes problem are presented

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

geometry and a priori estimates for free boundary problems of the euler s equation in this paper we derive estimates to the free boundary problem for the euler equation with surface tension and without surface tension provided the rayleigh taylor sign condition holds we prove that as the surface tension tends to 0 when the rayleigh taylor condition is satisfied solutions converge to the euler flow with zero surface tension 2007 wiley periodicals inc

a note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices in this note we consider a set of augmentation block preconditioners for solving generalized saddle point systems whose coefficient matrices have singular 1 1 blocks results concerning the eigenvalue distribution and forms of the eigenvectors of the augmentation block preconditioned generalized saddle point matrix and its minimal polynomial are given and an optimal augmentation block preconditioner of the set is derived these results extend previous ones in the literature numerical experiments that show the very effective performance of the optimal augmentation block preconditioner are reported

anisotropic decay and global well posedness of viscous surface waves without surface tension abstract we consider a viscous incompressible fluid below the air and above a fixed bottom the fluid dynamics is governed by the gravity driven incompressible navier stokes equations and the effect of surface tension is neglected on the free surface the global well posedness and long time behavior of solutions near equilibrium have been intriguing questions since beale 1981 2 it was proved by guo and tice 2013 6 that with certain additional low horizontal frequency assumption of the initial data in 3d an integrable decay rate of the velocity is obtained so that the global unique solution can be constructed while the global well posedness in 2d was left open in this paper by exploiting the anisotropic decay rates of the velocity which are even not integrable we prove the global well posedness in both 2d and 3d without any low frequency assumption of the initial data one of key observations here is a cancellation in nonlinear estimates of the viscous stress tensor term in the bulk by using alinhac good unknowns when estimating the energy evolution of the highest order horizontal spatial derivatives of the solution

a generalized relaxed positive definite and skew hermitian splitting preconditioner for non hermitian saddle point problems for non hermitian saddle point problems with the non hermitian positive definite 1 1 block zhang et al 2014 presented a relaxed positive definite and skew hermitian splitting rpss preconditioner to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the convergence property of the grpss iteration method is proved and a generalized rpss grpss preconditioner is proposed the grpss preconditioner is much closer to the coefficient matrix than the rpss preconditioner in certain norm which straightforwardly results in an grpss iteration method we employing the grpss preconditioner to accelerate some krylov subspace methods like gmres the spectral distribution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the grpss preconditioner