adaptive finite element methods for hyperbolic systems with application to transient acoustics the solution of the hyperbolic systems of equations governing transient acoustics by adaptive finite elements and various finite difference time discretization schemes is addressed emphasis is placed on the use of a class of implicit runge kutta methods for temporal approximations

regular multivariate sampling and approximation in lp shift invariant spaces abstract in this paper some results about regular multivariate generalized sampling in the l p setting 1 p are proven thus stable multivariate regular sampling formulas are derived for l p shift invariant spaces v p i e the l p closure of the linear span of the shifts of a finite set of generators these sampling formulas include regular samples of the function its derivatives and or some filtered versions of the function itself taken at a lattice of r d approximation schemes using these generalized sampling formulas are also included

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

a moving finite element method with error estimation and refinement for one dimensional time dependent partial differential equations we discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension the mesh is moved so as to equidistribute the spatial component of the discretization error in h 1 we present a method of estimating this error by using p hierarchic finite elements the error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement we discretize the partial differential equations in space using a galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error a system of ordinary differential equations for mesh velocities are used to control element motions we use existing software for stiff ordinary differential equations for the temporal integration of the solution the error estimate and the mesh motion computational results using a code based on our method are presented for several examples

differential polynomial expressions related to the kadomtsev petviashvili and korteweg de vries hierarchies an integrable nonlinear partial differential equation typically extends to a hierarchy of integrable equations there exist several recursive schemes for obtaining these hierarchies recently explicit expressions for the kdv hierarchy have been found we derive explicit expressions for the hierarchy associated with the kp equation the main tools are sato s theory hirota s formalism and bell s polynomials

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one dimensional partial differential equations pdes the particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by w huang y ren and r d russell 1994 siam j numer anal 31 709 we show how the accuracy of the computations is strongly dependent on the choice of monitor function and we present a monitor function that yields an optimal rate of convergence motivated by efficiency considerations for problems in two or more space dimensions we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical pde the accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time step control mechanism is integrated into the scheme

1 1 dimensional hamiltonian systems as symmetry constraints of the kadomtsev petviashvili equation the authors consider linear problems associated with the kadomtsev petviashvili equation they prove that the linear problems are 1 1 dimensional hamiltonian systems under the symmetry constraints moreover they find that the hamiltonian flows of the linear problems are commutative

an adaptive grid with directional control abstract a variational formulation of winslow s variable diffusion method is combined with a directional control functional to derive an adaptive grid generator for adaptive node movement the addition of directional control allows one to align mesh lines with a prescribed vector field while adapting the spacing to resolve the data it is shown that harmonic function theory can be applied to define the conditions under which unique solutions of the resulting elliptic equations exist the grid generator is applied to the complex problem of magnetic field reconnection driven by fluid instability the results demonstrate that the cost of a computation with an adaptive grid with directional control is only one tenth of the cost of either a calculation on an eulerian grid or on an adaptive grid without directional control

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

a matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling the aim of this work is to solve a question raised for average sampling in shift invariant spaces by using the well known matrix pencil theory in many common situations in sampling theory the available data are samples of some convolution operator acting on the function itself this leads to the problem of average sampling also known as generalized sampling in this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support thus low computational complexity is involved and truncation errors are avoided in practice it is accomplished by means of a fir filter bank an answer is given in the light of the generalized sampling theory by using the oversampling technique more samples than strictly necessary are used the original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which for a suitable choice of the sampling period becomes a matrix pencil this matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum moreover the optimality of the obtained solution is established

approximation from shift invariant subspaces of l 2 ℝ d a complete characterization is given of closed shift invariant subspaces of l2 rd which provide a specified approximation order when such a space is principal i e generated by a single function then this characterization is in terms of the fourier transform of the generator as a special case we obtain the classical strang fix conditions but without requiring the generating function to decay at infinity the approximation order of a general closed shift invariant space is shown to be already realized by a specifiable principal subspace

practical aspects of formulation and solution of moving mesh partial differential equations moving mesh partial differential equations mmpdes are used in the mmpde moving mesh method to generate adaptive moving meshes for the numerical solution of time dependent problems how mmpdes are formulated and solved is crucial to the efficiency and robustness of the method in this paper several practical aspects of formulating and solving mmpdes are studied they include spatial balance scaling invariance effective control of mesh concentration bounds on time steps multiple sub steps and two level mesh movement numerical results are also given c 2001 academic press

a posteriori error analysis for numerical approximations of friedrichs systems the global error of numerical approximations for symmetric positive systems in the sense of friedrichs is decomposed into a locally created part and a propagating component residual based two sided local a posteriori error bounds are derived for the locally created part of the global error these suggest taking the l 2 norm as well as weaker dual norms of the computable residual as local error indicators the dual graph norm of the residual vec r h is further bounded from above and below in terms of the l 2 norm of h vec r h where h is the local mesh size the theoretical results are illustrated by a series of numerical experiments

oversampled filter banks perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in l sup 2 z these frames are the subject of this paper first the necessary and sufficient conditions of a filter bank for implementing a frame or a tight frame expansion are established as well as a necessary and sufficient condition for perfect reconstruction using fir filters after an fir analysis complete parameterizations of oversampled filter banks satisfying these conditions are given further we study the condition under which the frame dual to the frame associated with an fir filter bank is also fir and give a parameterization of a class of filter banks satisfying this property then we focus on non subsampled filter banks non subsampled filter banks implement transforms similar to continuous time transforms and allow for very flexible design we investigate the relations of these filter banks to continuous time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two channel filter banks that yield highly regular wavelets with a given number of vanishing moments

the kaup broer system and trilinear odes of painleve type the kaup broer hierarchy can be considered as a certain reduction of the kp hierarchy upon introducing the function the kaup broer system can be formulated in trilinear form giving rise to wronskian solutions in this paper several odes of the painleve type obtained through reductions of the kaup broer system are considered those odes take a trilinear form which allows a linearization

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

an error indicator monitor function for an r adaptive finite element method an r adaptive finite element method based on moving mesh partial differential equations pdes and an error indicator is presented the error indicator is obtained by applying a technique developed by bank and weiser to elliptic equations which result in this case from temporal discretization of the underlying physical pdes on moving meshes the construction of the monitor function based on the error indicator is discussed numerical results obtained with the current method and the commonly used method based on solution gradients are presented and analyzed for several examples c 2001 academic press

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

asymmetric multi channel sampling in shift invariant spaces abstract we develop an asymmetric multi channel sampling on a shift invariant space v with a riesz generator t in l 2 r where each channeled signal is assigned a uniform but distinct sampling rate we use fourier duality between v and l 2 0 2 to find conditions under which there is a stable asymmetric multi channel sampling formula on v

recovery of missing samples for oversampling in shift invariant spaces abstract it is well known that any finite missing samples of a band limited signal can be recovered from the remaining known samples when the signal is oversampled at a rate higher than the minimum nyquist rate we consider a similar problem for oversampled signals in shift invariant spaces with compactly supported continuous riesz generators using a multi channel oversampling we find conditions under which finitely many missing samples can be recovered

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

robust a posteriori error estimators for a singularly perturbed reaction diffusion equation we derive robust a posteriori error estimators for a singularly perturbed reaction diffusion equation here robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter the estimators are based either on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

movcol4 a moving mesh code for fourth order time dependent partial differential equations in this paper we develop and analyze a moving mesh code for the adaptive simulation of fourth order pdes based on collocation the scheme is shown to enforce discrete conservation for problems written in a generalized conservation form to demonstrate the breadth of applicability we present examples from both cahn hilliard and thin film type equations exhibiting metastable behavior finite time solution blow up finite time extinction and moving interfaces

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

darboux theorems and wronskian formulas for integrable systems i constrained kp flows abstract generalizations of the classical darboux theorem are established for pseudo differential scattering operators of the form l i 0 n u i i i 1 m i 1 i i iteration of the darboux transformations leads to a gauge transformed operator with coefficients given by wronskian formulas involving a set of eigenfunctions of l nonlinear integrable partial differential equations are associated with the scattering operator l which arise as a symmetry reduction of the multicomponent kp hierarchy with a suitable linear time evolution for the eigenfunctions the darboux transformation is used to obtain solutions of the integrable equations in terms of wronskian determinants

design and application of a gradient weighted moving finite element code ii in two dimensions in part i the authors reported on the design of a robust and versatile gradient weighted moving finite element gwmfe code in one dimension and on its application to a variety of pdes and pde systems this companion paper does the same for the two dimensional 2d case these moving node methods are especially suited to problems which develop sharp moving fronts especially problems where one needs to resolve the fine scale structure of the fronts the many potential pitfalls in the design of gwmfe codes and the special features of the implicit one dimensional 1d and 2d codes which contribute to their robustness and efficiency are discussed at length in part i this paper concentrates on issues unique to the 2d case brief explanations are given of the variational interpretation of gwmfe the geometrical mechanical interpretation simplified regularization terms and the treatment of systems a catalog of inner products which occur in gwmfe is given with particular attention paid to those involving second order operators after presenting an example of the 2d phenomenon of grid collapse and discussing the need for long time regularization the paper reports on the application of the 2d code to several nontrivial problems nonlinear arsenic diffusion in the manufacture of semiconductors the drift diffusion equations for semiconductor device simulation the buckley leverett black oil equations for reservoir simulation and the motion of surfaces by mean curvature

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

review of some adaptive node movement techniques in finite element and finite difference solutions of partial differential equations abstract this article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations the methods discussed involve the adaptive movement of nodes so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables surrounded by regions of relatively smooth variation flows with shock waves contact surfaces slip streams phase change interfaces and boundary layers can be modelled with great precision by these methods it will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables

adaptive finite element methods for parabolic problems v long time integration we continue our previous work on adaptive finite element methods for parabolic problems now with particular emphasis on long time integration for semidefinite problems

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

oversampling and reconstruction functions with compact support assume that a sequence of samples of a filtered version of a function in a shift invariant space is given this paper deals with the existence of a sampling formula involving these samples and having reconstruction functions with compact support this is done in the light of the generalized sampling theory by using the oversampling technique a necessary and sufficient condition is given in terms of the smith canonical form of a polynomial matrix finally we prove that the aforesaid oversampled formulas provide nice approximation schemes with respect to the uniform norm

on a uniformly accurate finite difference approximation of a singulary peturbed reaction diffusion problem using grid equidistribution we examine the convergence properties of a finite difference approximation of a singularly perturbed reaction diffusion boundary value problem using a nonuniform grid the grid is based on the equidistribution of a positive monitor function that is a linear combination of a constant floor and a power of the second derivative of the solution analysis shows how the monitor function can be chosen to ensure that the accuracy of the numerical approximation is insensitive to the size of the singular perturbation parameter the use of equidistribution principles appears in many practical grid adaption schemes and our analysis provides insight into the convergence behaviour on such grids numerical results are given that confirm the uniform convergence rates

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

constraints of the kadomtsev petviashvili hierarchy for the kadomtsev petviashvili kp hierarchy constructed in terms of the famous sato theory a k constraint is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates the eigenvalue problem of the k constrained system is naturally obtained from the linear system of the kp hierarchy which takes the form of kth order polynomial coupled with a first order one thus we are able to derive the correspondent lax pair recursion operator bi hamiltonian structures and conserved quantities the constraints for the bkp hierarchy are also sketched

a posteriori error estimates based on hierarchical bases the authors present an analysis of an a posteriors error estimator based on the use of hierarchical basis functions the authors analyze nonlinear nonselfadjoint and indefinite problems as well as the selfadjoint positive definite case because both the analysis and the estimator itself are quite simple it is easy to see how various approximations affect the quality of the estimator as examples the authors apply the theory to some scalar elliptic equations and the stokes system of equations

generalized sampling theorems in multiresolution subspaces it is well known that under very mild conditions on the scaling function multiresolution subspaces are reproducing kernel hilbert spaces rkhss this allows for the development of a sampling theory in this paper we extend the existing sampling theory for wavelet subspaces in several directions we consider periodically nonuniform sampling sampling of a function and its derivatives oversampling multiband sampling and reconstruction from local averages all these problems are treated in a unified way using the perfect reconstruction pr filter bank theory we give conditions for stable reconstructions in each of these cases sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported except in trivial cases this restriction no longer applies for the generalizations we study here due to the existence of fir pr banks in fact with nonuniform sampling oversampling and reconstruction from local averages we can guarantee compactly supported synthesizing functions moreover local averaging schemes have additional nice properties robustness to the input noise and compression capabilities we also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis after this we extend the sampling theory to random processes we require autocorrelation functions to belong to some subspace related to wavelet subspaces it turns out that we cannot recover random processes themselves unless they are bandlimited but only their power spectral density functions we consider both uniform and nonuniform sampling

oversampling and aliasing in de branges spaces arising from bessel operators abstract we show that a class of de branges spaces generated by means of generalized fourier transforms associated with perturbed bessel differential equations has the properties of oversampling and aliasing

a study of monitor functions for two dimensional adaptive mesh generation in this paper we study the problem of two dimensional adaptive mesh generation using a variational approach and specifically the effect that the monitor function has on the resulting mesh behavior the basic theoretical tools employed are green s function for elliptic problems and the eigendecomposition of symmetric positive definite matrices based upon this study a general strategy is suggested for how to choose the monitor function and numerical results are presented for illustrative purposes the three dimensional case is also briefly discussed it is noted that the strategy used here can be applied to other elliptic mesh generation techniques as well

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

constraints of the kp hierarchy and multilinear forms we consider the trilinear form of the kaup broer system which gives rise to solutions in wronskian form the kaup broer system is connected with akns system through a gauge transformation the akns hierarchy can be understood as a generalized 1 constraint of the kp hierarchy imposing that constraint on sato s equation we obtain the basic trilinear form and moreover a hierarchy of trilinear equations governing the akns flows similary hierarchies of multilinear forms are derived in the case of generalized k constraints