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

on the index for array optimization and the discrete prolate spheroidal functions a class of array optimization problems is considered in which we seek to optimize the array response in a specified angular sector the optimization of array directivity is shown to be a special limiting case of these problems as the width of the specified angular sector approaches zero the optimum array patterns are also shown to be related to the well known prolate spheroidal functions

spatial spectral filtering with linearly constrained minimum variance beamformers in this paper orthogonal low dimension models for broad band sources as observed by an array delay line structure are developed the models are termed broad band source representation spaces and are subspaces of the observation space they are generated from eigenstructures of source sample covariance matrices and represent sources efficiently in the 2nd order statistical sense for a single broad band source propagating in a pure delay environment and observed with an isotropic array an accurate indication of the model dimension is derived which is independent on the array configuration being a function of the observed source time bandwidth product only using these representations procedures are described for controlling the spatial and spectral response of linearly constrained minimum variance broad band array beamformers the procedures employ linear constraints to control complex gain characteristics over selected bands of frequency for multiple points of 3 d regions of source locations and do not require the use of steering delays the constraints are derived from source representation spaces and are termed eigenvector constraints simulations are presented which illustrate the effectiveness of the new linear constraints

on the rate of convergence for the approximation of nonlinear problems in this paper we present an abstract theory to obtain error estimates in various norms for the approximation of solution branches of nonlinear equations by the aid of known estimates for corresponding linear problems

convergence cones near bifurcation the performance of predictor solver continuation methods near bifurcation points is considered a general theory is developed for the convergence of two commonly used solvers namely newton s method and the chord method near singular points of a nonlinear equation this theory provides a uniform treatment of continuation for simple bifurcation multiple bifurcation and multiple limit point bifurcation problems for these types of bifurcation it is shown that there are conical domains of attraction for the above iterative methods centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations

hp adaptive discontinuous galerkin methods for bifurcation phenomena in open flows we consider the application of high order hp version adaptive discontinuous galerkin finite element methods for the discretization of the bifurcation problem associated with the steady incompressible navier stokes equations based on exploiting the dual weighted residual approach reliable and efficient a posteriori estimates of the error in the computed critical reynolds number at which a steady pitchfork bifurcation occurs when the underlying physical system possesses either reflectional z 2 symmetry or rotational and reflectional o 2 symmetry are derived numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on hp adaptively refined computational meshes are presented for both two and three dimensional problems

how to get around a simple quadratic fold we analyse from a theoretical point of view a computational technique previously introduced 2 5 for tracing branches of solutions of nonlinear equations near simple quadratic folds

optimal homotopy methods for solving nonlinear systems this paper deals with the monotone homotopy methods for solving the system of nonlinear equations in sect 2 the homotopy with a distancemonotone homotopy path is discussed and it is proved that a homotopy with a given structure has a distance monotone homotopy path under some regular conditions then two structure variable homotopy algorithms called local straighten up method and global straighten up method are developed to approximate the optimal homotopy and an adaptive step size control strategy for these algorithms is proposed respectively a new method for analyzing the convergence is presented which is based on the geometrical properties of the algorithm then the convergence of the algorithm is proved under certain regular conditions finally two numerical examples are given to illustrate the effectiveness of the above two algorithms

the numerical solution of nonlinear equations having several parameters part iii equations with z 2 symmetry the computation of symmetry breaking bifurcation points of nonlinear multiparameter problems with z 2 reflectional symmetry is considered the numerical approach is based on recent work in singularity theory which is used to construct systems of equations and inequalities characterising various types of symmetry breaking bifurcation points numerical continuation methods are then used to follow paths of symmetry breaking bifurcations and hence compute regions in parameter space for which a problem has qualitatively similar bifurcation diagrams the power of the numerical approach is illustrated by computations of axisymmetric flows in the finite taylor problem

an analysis of upwind schemes for the navier stokes equations we state a general result on approximation of branches of nonsingular solutions of nonlinear problems the abstract theory is then applied to the study of two mixed finite element schemes for the navier stokes equations using an upwind discretization of the convective terms

a priori error estimates of finite element solutions of parametrized strongly nonlinear boundary value problems abstract nonlinear boundary value problems with parameters are called parametrized nonlinear boundary problems this paper studies a priori error estimates of finite element solutions of second order parametrized strongly nonlinear boundary value problems in divergence form on one dimensional bounded intervals the banach space w 0 1 is chosen in formulation of the error analysis so that the nonlinear differential operators defined by the differential equations are nonlinear fredholm operators of index 1 finite element solutions are defined in a natural way and several a priori estimates are proved on regular branches and on branches around turning points in the proofs the extended implicit function theorem due to brezzi et al 1980 plays an essential role

optimal order error estimates for finite element approximations of a bifurcation function the bifurcation function associated with an elliptic boundary value problem au g u 0 is a vector field b on rd with the property that the solutions of the boundary value problem are in a one to one correspondence with the zeros of b a finite element approximation bh of b is formulated and optimal order error estimates are derived implementation issues are also discussed

finite dimensional approximation of bifurcation problems in presence of symmetries we give here a brief summary of the results of 4 14 the aim of our work was not only the finite dimensional approximation of the specific problem studied in 3 but we also wanted to show how to adapt to discrete problems and how to use in this case some classical mathematical tools e g the splitting lemma of gromoll meyer magnus the singularity theory etc such considerations are developed in 5 in particular we prove in 5 that in most cases e g in the examples given in 6 7 9 10 the discrete bifurcation problem behaves as an imperfect bifurcation problem and we improve the methods of 12 another way to use universal unfoldings in the approxima tion of bifurcation problems is suggested by 2

on the calculation of limit and bifurcation points in stability problems of elastic shells abstract numerical methods for the computation of singular points of nonlinear equations g u 0 are discussed where and are real parameters simple and double limit points are treated in some detail and numerical algorithms are presented and applied to elastic shell stability problems the case of simple symmetry breaking bifurcation points is also treated with applications to nonsymmetric bifurcation from axisymmetric states of deformation of shells of revolution

a c 0 c0 interior penalty method for a von karman plate we investigate a quadratic c 0 c0 interior penalty method for the approximation of isolated solutions of a von karman plate we prove that the discrete problem is uniquely solvable near an isolated solution and establish optimal order error estimates numerical results that illustrate the theoretical estimates are also presented

a conforming finite element method for the time dependent navier stokes equations we study a spatial discretization of the stokes problem in a domain of mathbb r 2 or mathbb r 3 by a finite element method the unknowns being the velocity and the pressure optimal error

numerical solution of the three dimensional ginzburg landau models using artificial boundary for three dimensional simulations of the vortex phenomena in superconductors based on the ginzburg landau gl theory the physical variables must be solved in the whole space in general exact boundary conditions on an artificial boundary are discussed to reformulate the problem in a finite domain approximations of these boundary conditions with high accuracy are given and their convergence properties are examined error estimates are derived for the finite element approximations with the approximate boundary conditions

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

an optimization based domain decomposition method for pdes with random inputs an optimization based domain decomposition method for stochastic elliptic partial differential equations is presented the main idea of the method is a constrained optimization problem for which the minimization of an appropriate functional forces the solutions on the two subdomains to agree on the interface the constraints are the stochastic partial differential equations the existence of optimal solutions for the stochastic optimal control problem is shown as is the convergence to the exact solution of the given problem we prove the existence of a lagrange multiplier and derive an optimality system from which solutions of the domain decomposition problem may be determined finite element approximations to the solutions of the optimality system are defined and analyzed with respect to both spatial and random parameter spaces then the results of some numerical experiments are given to confirm theoretical error estimate results

computational methods for bifurcation problems with symmetries with special attention to steady state and hopf bifurcation points we show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations we extend the approach by werner 1988 by presenting methods for the detection of bifurcation points and the computation of multiple hopf points the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group the group theoretical tool is an a priori knowledge of the interaction of certain subgroups 0 and of at possibly multiple steady state or hopf bifurcation points minimal 0 breaking bifurcation we introduce a bifurcation graph which shows graphically this a priori information its edges represent possible symmetry breaking bifurcations our analysis follows the lines of golubitsky stewart and schaeffer 1988 but it is aimed to numerical applications we have chosen a 4 box brusselator model in order to explain our notions and ideas and to discuss the numerical procedure

a posteriori error estimators for the stokes equations ii non conforming discretizations we present an a posteriori error estimator for the non conforming crouzeix raviart discretization of the stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation the error estimator yields global upper and local lower bounds for the error of the finite element solution it can easily be generalized to the stationary incompressible navier stokes equations and to other non conforming finite element methods numerical examples show the efficiency of the proposed error estimator

une methode numerique pour le calcul des points de retournement application a un probleme aux limites non lineaire an iterative method is presented for determining a turning point in a branch of solutions of an algebraic system of equations depending on a parameter one shows that the convergence is superlinear and that this method may constitute a complement for the newton method with a finite element method numerical results are given testing the process on the discrete version of a mildly nonlinear boundary value problem to determine the turning point in the positive solution branch

computational error estimates and adaptive processes for some nonlinear structural problems abstract in earlier papers the authors introduced and analyzed the calculation of reliable a posteriori error estimates for finite element solutions and discussed the design and effectivity of adaptive procedures based upon them while most of this work concerned linear problems this paper is intended to show that the same approaches remain also highly effective in the nonlinear case as a model problem a planar elastic rod is considered involving both material as well as geometrical nonlinearities but as far as possible the discussion is kept general for nonlinear problems of this type interest centers on an analysis of the shape and features of the equilibrium surface after some general remarks about such surfaces and the related computational problems a general continuation process is summarized and some of its extensions for determining limit points and tracing stability paths are discussed then estimators are introduced for the error along the solution paths and of the computed critical values experimental results for the model problem show the effectivity of these estimators and of the adaptive procedures based upon them then some aspects relating to the occurrence and identification of spurious solutions are discussed in order to illustrate desirable generalizations of these results the paper ends with an outline of some results for linear problems of the type that should be achievable also in the nonlinear case

analysis of an interface relaxation method for composite elliptic differential equations the theoretical analysis on both the continuous differential and the discrete linear algebra levels of an interface relaxation method for solving elliptic differential equations is presented the convergence of the method for 1 dimensional problems is proved the region of convergence and the optimal values for the relaxation parameters involved are determined for model problems numerical data for 1 and 2 dimensional problems that confirm the theoretical results exhibit the effectiveness of the method and elucidate its characteristics are presented

finite dimensional approximation of nonlinear problems part ii limit points note univ paris 6 f 75230 paris 05 france ecole polytech ctr math appl cnrs equipe rech 747 f 91128 palaiseau france univ pavia cnr ist anal numer i 27100 pavia italy brezzi f univ pavia cnr ist math appl i 27100 pavia italy isi document delivery no lr761times cited 54cited reference count 13 reference asn article 1981 002doi 10 1007 bf01396184 record created on 2006 08 24 modified on 2017 05 12

a posteriori existence in numerical computations a typical a posteriori error analysis for numerical methods for differential equations consists of two elements the stability of the continuous problem and an a posteriori computed residual estimate if the problem is nonlinear then this is possible only if it is strongly monotone in this paper we will estimate the stability constant a posteriori as well in order to apply the inverse function theorem to derive the existence of solutions from the numerical computation and thus obtain rigorous a posteriori error estimates for nonmonotone equations

the discrete prolate spheroidal filter as a digital signal processing tool the discrete prolate spheroidal dps filter is one of the class of nonrecursive finite impulse response fir filters the dps filter first introduced by tufts and francis 1 is superior to other filters in this class in that it has maximum energy concentration in the frequency passband and minimum ringing in the time domain slepian 2 gives a complete discussion of dps function properties we give a mathematical development of the dps filter properties provide information required to easily and accurately construct even many coefficient filters and compare the properties of this filter with those of the more commonly used filters of the same class we note that use of the dps filter allows for particularly useful statements of data time frequency resolution cell values and that overall it forms an especially useful though little known tool for digital signal processing

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

forced oscillation of a class of neutral hyperbolic differential equations in this paper we study the following boundary value problem for a class of neutral hyperbolic differential equations 2 t2 u c t u x t a0 t u a1 t u x t ab q x t f u x g t d g x t x t r g u n v x t u 0 x t r a number of theorems for oscillatory solutions of the problem under two different cases are developed

numerical solution to a mixed navier stokes darcy model by the two grid approach we study numerical methods for a coupled navier stokes darcy model which describes a fluid flow filtrating through porous media a decoupled and linearized two grid algorithm is proposed numerical analysis and experiments are presented to show the efficiency and effectiveness of the decoupled and linearized algorithm

multiple cluster beamspace and resolution enhanced esprit this paper addresses approaches to enhancement of the resolution of one or more clusters of closely spaced emitters the tls esprit algorithm is considered as applied in beamspace and in element space in conjunction with resolution enhancement beamspace bs esprit employs a matrix beamformer as a preprocessor to map the sensor space into a lower dimensional beamspace resolution enhanced re esprit employs a matrix beamformer as a left weighting of the signal subspace eigenvector matrix for enhancing the resolution of esprit this paper specifically discusses several types of these beamformer matrices which differ in the way that they treat source clusters other than of interest our objective is to clearly define approaches and identify their relative merits through discussion and illustrative simulation so as to provide an understanding of how to proceed in designing an esprit algorithm

finite element simulations of window josephson junctions this paper deals with the numerical simulation of the steady state two dimensional window josephson junctions by finite element method the model is represented by a sine gordon type composite pde problem convergence and error analysis of the finite element approximation for this semilinear problem are presented an efficient and reliable newton preconditioned conjugate gradient algorithm is proposed to solve the resulting nonlinear discrete system regular solution branches are computed using a simple continuation scheme numerical results associated with interesting physical phenomena are reported interface relaxation methods which by taking advantage of special properties of the composite pde can further reduce the overall computational cost are proposed the implementation and the associated numerical experiments of a particular interface relaxation scheme are also presented and discussed

on the theory and error estimation of the reduced basis method for multi parameter problems abstract in an earlier paper zamm 63 1983 21 j p fink and w c rheinboldt developed a priori local error estimates for the scalar parameter case of the reduced basis method by considering the method in a differential geometric setting here it is shown that an analogous setting can be used for the analysis of the method applied to problems with a multidimensional parameter vector and that this leads to a corresponding local error theory also in this general case

an optimization based domain decomposition method for the navier stokes equations an optimization based nonoverlapping domain decomposition method for the solution of the navier stokes equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the navier stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces we show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined finite element approximations of the solutions of the optimality system are examined the domain decomposition method is also reformulated as a nonlinear least squares problem and the results of some numerical experiments are given

oscillation criteria of nonlinear hyperbolic equations with functional arguments abstract in this paper we consider a class of hyperbolic equations with functional arguments and obtain sufficient conditions for the oscillation of solutions of the equations with two kinds of boundary conditions

semidiscrete approximations of optimal robin boundary control problems constrained by semilinear parabolic pde an optimal robin boundary control problem associated with semilinear parabolic partial differential equations is considered existence of an optimal solution is proved and an optimality system of equations is derived semidiscrete finite element approximations of the optimality system are defined and error estimates are obtained

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

approximation of hopf bifurcation we make several assumptions on a nonlinear evolution problem ensuring the existence of a hopf bifurcation under a fairly general approximation condition we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions

analysis of the l2 least squares finite element method for the velocity vorticity pressure stokes equations with velocity boundary conditions a theoretical analysis of the l 2 least squares finite element method lsfem for solving the stokes equations in the velocity vorticity pressure vvp first order system formulation with the dirichlet velocity boundary condition is given the least squares energy functional is defined to be the sum of the squared l 2 norms of the residuals in the partial differential equations weighted appropriately by the viscosity constant n it is shown that with many advantages the method is stable and convergent without requiring extra smoothness of the exact solution and the piecewise linear finite elements can be used to approximate all the unknowns furthermore with respect to the order of approximation for smooth exact solutions the method exhibits an optimal rate of convergence in the h 1 norm for velocity and a suboptimal rate of convergence in the l 2 norm for vorticity and pressure some numerical experiments in two and three dimensions are given which confirm the a priori error estimates since the boundary of the bounded domain under consideration is polygonal in r 2 or polyhedral in r 3 instead of c 1 smooth the authors adopt the more direct technique of bramble pasciak and cai manteuffel mccormick which departs from the agmon douglis nirenberg theory in showing the coercivity bound of the least squares functional

primal mixed finite element approximation of elliptic equations with gradient nonlinearities we study the primal mixed finite element approximation of the second order elliptic problem with gradient nonlinearities existence and uniqueness of the approximate solution are proved and optimal order a priori error estimates are obtained also reliable a posteriori error estimators are derived

large sparse continuation problems we study and develop efficient and versatile predictor corrector continuation methods for large sparse problems the first object is to show how special solving methods for a large sparse linear systems can be incorporated into the basic steps of a continuation method next we describe how to use a special nonlinear conjugate gradient method to perform the corrector phase it is shown how such methods can be used to detect bifurcation points and how to trace bifurcating solution branches by using local perturbations finally a numerical example involving bifurcating branches of a nonlinear eigenvalue problem is given

solving composite problems with interface relaxation this paper deals with a solution approach suitable for composite partial differential equations pdes with interface conditions we present a general framework based on interface relaxation which provides a uniform platform for building problem solving environments through efficient software integration and for implementing various relaxation schemes mathematically this framework contains many existing domain decomposition methods and also allows the extension to a variety of new relaxers in particular we describe a class of relaxers which are suitable for general and complicated interface conditions interface relaxation is more general than the traditional domain decomposition methods in that it allows unrelated pde problems on different subdomains convergence analysis error estimates and preconditioning strategies are presented which show that these relaxers are competitive with existing domain decomposition methods for model problems involving a single pde we present experimental results which demonstrate the wide applicability of this approach differences between this approach and other domain decomposition methods are also discussed

computation and stability of fluxons in a singularly perturbed sine gordon model of the josephson junction new numerical results are reported regarding the solutions of a singularly perturbed sine gordon equation modeling magnetic flux quanta fluxons in long josephson tunnel junctions with nonzero surface impedance previous authors hampered by an inadequate numerical shooting method conjectured that the fluxon branch of solitary wave solutions terminates at a critical value of the bias current in the present paper a new numerical boundary value solver is developed and then used to show that the critical value of the bias current corresponds not to a termination point but to a turning point in the bifurcation diagram plotting a suitable norm of the solution against the bias current we find that the solution curve turns back at the critical value and then oscillates via a sequence of turning points to a limiting value of the bias current below the value at the first turning point these conclusions are confirmed a posteriors by adapting current numerical procedures described in the literature e g

oscillation criteria for boundary value problems of high order partial functional differential equations in this paper a class of boundary value problems associated with high order partial functional differential equations with distributed deviating arguments is investigated some oscillation criteria of solutions to the problem are developed our approach is to reduce the multi dimensional oscillation problem to a one dimensional oscillation one by employing some integral means of solutions and introducing some parameter functions one illustrative example is considered

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions