parameter uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction diffusion problems abstract we consider a dirichlet boundary value problem for singularly perturbed semilinear reaction diffusion equation the problem is discretized using an exponential spline difference scheme derived on the basis of splines in tension on piecewise uniform shishkin type mesh the convergence analysis is given and the method is shown to be almost second order accurate in the discrete maximum norm uniformly in the perturbation parameter e numerical experiments are conducted to demonstrate the theoretical results

some upwinding techniques for finite element approximations of convection diffusion equations a uniform framework for the study of upwinding schemes is developed the standard finite element galerkin discretization is chosen as the reference discretization and differences between other discretization schemes and the reference are written as artificial diffusion terms these artificial diffusion terms are spanned by a four dimensional space of element diffusion matrices three basis matrices are symmetric rank one diffusion operators associated with the edges of the triangle the fourth basis matrix is skew symmetric and is associated with a rotation by 2 while finite volume discretizations may be written as upwinded galerkin methods the converse does not appear to be true our approach is used to examine several upwinding schemes including the streamline diffusion method the box method the scharfetter gummel discretization and a divergence free scheme

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

finite volume solutions of convection diffusion test problems the cell vertex formulation of the finite volume method has been developed and widely used to model inviscid flows in aerodynamics more recently one of us has proposed an extension for viscous flows the purpose of the present paper is two fold first we have applied this scheme to a well known convection diffusion model problem involving flow round a 180 degrees bend which highlights some of the issues concerning the application of the boundary conditions in such cell based schemes the results are remarkably good when the boundary conditions are applied in an appropriate manner in our efforts to explain the high quality of the results we were led to a detailed analysis of the corresponding one dimensional problem our second purpose is thus to gather together various approaches to the analysis of this problem and to draw attention to the supra convergence phenomena enjoyed by the proposed methods 25 refs 9 figs 1 tab

a globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions we analyze a new galerkin finite element method for numerically solving a linear convection dominated convection diffusion problem in two dimensions the method is shown to be convergent uniformly in the perturbation parameter of order h1 2 in a global energy norm which is stronger than the l2 norm this order is optimal in this norm for our choice of trial functions

on the non existence of e uniform finite difference methods on uniform meshes for semilinear two point boundary value problems in this paper fitted finite difference methods on a uniform mesh with internodal spacing h are considered for a singularly perturbed semilinear two point boundary value problem it is proved that a scheme of this type with a frozen fitting factor cannot converge e uniformly in the maximum norm to the solution of the differential equation as the mesh spacing h goes to zero numerical experiments are presented which show that the same result is true for a number of schemes with variable fitting factors

the finite difference streamline diffusion methods for sobolev equations with convection dominated term in this paper we give two finite difference streamline diffusion fdsd schemes for solving linear sobolev equations with convection dominated term stability and optimal error estimates for them are derived in suitable norms

tree compression with top trees we introduce a new compression scheme for labeled trees based on top trees our compression scheme is the first to simultaneously take advantage of internal repeats in the tree as opposed to the classical dag compression that only exploits rooted subtree repeats while also supporting fast navigational queries directly on the compressed representation we show that the new compression scheme achieves close to optimal worst case compression can compress exponentially better than dag compression is never much worse than dag compression and supports navigational queries in logarithmic time

the solution of singularly perturbed parabolic problems by parallel algorithms combining crank nicholson scheme and overlapping schwarz methods abstract parallel algorithms combining a time discretization and overlapping domain decomposition methods are applied to the solution of singularly perturbed parabolic problems two domain decomposition methods based on the schwarz alternating procedure are proposed the method with two colour ordering of subdomains and the method with additional correcting problems moreover modifications of the methods are considered using time extrapolation on subdomain interfaces convergence properties of the algorithms are established at the differential level numerical results for a test singularly perturbed problem are presented to confirm the convergence theory and to evaluate the parallel arithmetic complexity of the algorithms

the smallest grammar problem this paper addresses the smallest grammar problem what is the smallest context free grammar that generates exactly one given string spl sigma this is a natural question about a fundamental object connected to many fields such as data compression kolmogorov complexity pattern identification and addition chains due to the problem s inherent complexity our objective is to find an approximation algorithm which finds a small grammar for the input string we focus attention on the approximation ratio of the algorithm and implicitly the worst case behavior to establish provable performance guarantees and to address shortcomings in the classical measure of redundancy in the literature our first results are concern the hardness of approximating the smallest grammar problem most notably we show that every efficient algorithm for the smallest grammar problem has approximation ratio at least 8569 8568 unless p np we then bound approximation ratios for several of the best known grammar based compression algorithms including lz78 b isection sequential longest match greedy and re pair among these the best upper bound we show is o n sup 1 2 we finish by presenting two novel algorithms with exponentially better ratios of o log sup 3 n and o log n m sup where m sup is the size of the smallest grammar for that input the latter algorithm highlights a connection between grammar based compression and lz77

a velocity pressure streamline diffusion finite element method for navier stokes equations a streamline diffusion finite element method is introduced for the time dependent incompressible navier stokes equations in a bounded domain in r squared and r cubed in the case of a flow with a high reynolds number an error estimate is proved and numerical results are given the method is based on a mixed velocity pressure formulation using the same finite element discretization of space time for the velocity and the pressure spaces which consist of piecewise linear functions together with certain least squares modifications of the galerkin variational formulation giving added stability without sacrificing accuracy

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

a uniformly accurate finite element method for a singularly perturbed one dimensional reaction diffusion application d une methode a elements finis avec des elements de base exponentiels a un probleme autoadjoint a perturbation singuliere et a deux points limites generation d un schema aux differences tridiagonal uniformement precis du second ordre

error estimates for a class of partial functional differential equation with small dissipation this article deals with a class of partial functional differential equation with a small dissipating parameter on a rectangular domain classical numerical methods for solving this type of problems reveal disappointing behavior or are tremendously expensive in computer memory and processor time this arises because the precision of an approximate solution depends inversely on perturbation parameter values and thus it deteriorates as a parameter decreases therefore it is of particular interest to develop numerical methods whose error estimates would be independent of the perturbation parameter contaminating the solution in order to overcome the said difficulty we derive robust parameter uniform error estimates for a class of partial functional differential equations the analysis presented in this paper uses a suitable decomposition of the error into smooth and a singular component combined with the appropriate barrier functions and comparison principle

fourier transform vector quantization for speech coding design algorithms and simulation results are presented for vector quantizers for fourier transformed data transforming the data prior to quantization has two potential advantages first each sample in the transform domain depends on many samples in the original domain thus even scalar quantization in the transform domain is a form of vector quantization or block source coding in the original waveform domain and the basic coding theorems of information theory show that such block codes can provide better performance than scalar codes even for memoryless sources second vector quantization of fourier transformed speech waveforms provides distinctly better subjective quality than ordinary vector quantization of the waveform using codes of comparable complexity while the system is of course more complicated due to the need to take fourier transforms its envisioned application is as a coder for the output of fft chips currently available or under development the proposed implementation of a fourier transform vector quantizer ftvq uses a product code structure providing different codes for different coefficient vectors corresponding to different frequency bands this is a form of subband coding and yields a simple means of optimizing bit allocations among the subcodes two coding structures with corresponding distortion measures are considered those that quantize vectors of pairs of real and imaginary coefficients and those that quantize separate vectors of magnitude and phase coefficients both structures yield good performance for the given complexity in comparison to waveform vector quantizers for speech coding a magnitude phase ftvq yields better subjective quality than a real imaginary ftvq when the rate allocation is properly chosen

fitted mesh b spline collocation method for solving self adjoint singularly perturbed boundary value problems in this paper we develop b spline collocation method for solving a class of self adjoint singularly perturbed equations given by 0 1 email protected a x y b x y x g x a x a 0 a x 0 b x b 0 y 0 email protected y 1 email protected use the fitted mesh technique to generate piecewise uniform mesh and use b spline method which leads to a tridiagonal linear system the convergence analysis is given and the method is shown to have uniform convergence of second order numerical illustrations are given in the end to demonstrate the efficiency of our method

a modified diffusion coefficient technique for the convection diffusion equation a new modified diffusion coefficient mdc technique for solving convection diffusion equation is proposed the galerkin finite element discretization process is applied on the modified equation rather than the original one for a class of one dimensional convection diffusion equations we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size then prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and or equation coefficients the application of the derived analytic formula of mdc is extended for other classes of convection diffusion equations where the analytic formula is computed locally within each element according to the mesh size and the values of associated coefficients in each direction the numerical results of the proposed approach for two dimensional variable coefficients with boundary layers convection dominated problems show stable and accurate solutions even on coarse grids accordingly multigrid based solvers retain their efficient convergence rates

space efficient preprocessing schemes for range minimum queries on static arrays given a static array of n totally ordered objects the range minimum query problem is to build a data structure that allows us to answer efficiently subsequent on line queries of the form what is the position of a minimum element in the subarray ranging from i to j we focus on two settings where 1 the input array is available at query time and 2 the input array is available only at construction time in setting 1 we show new data structures a of size frac 2n c n theta bigl frac n lg lg n c n lg n bigr bits and query time o c n for any positive integer function c n in o bigl n varepsilon bigr for an arbitrary constant 0 varepsilon 1 or b with o nh k o n bits and o 1 query time where h k denotes the empirical entropy of k th order of the input array in setting 2 we give a data structure of size 2n o n bits and query time o 1 all data structures can be constructed in linear time and almost in place

on supraconvergence phenomenon for second order centered finite differences on non uniform grids in the present note we consider an example of a boundary value problem for a simple second order ordinary differential equation which may exhibit a boundary layer phenomenon we show that usual central finite differences which are second order accurate on a uniform grid can be substantially upgraded to the fourth order by a suitable choice of the underlying non uniform grid this example is quite pedagogical and may give some ideas for more complex problems

the finite difference streamline diffusion method for the incompressible navier stokes equations in this paper a finite difference streamline diffusion scheme is introduced for solving the incompressible navier stokes equations for the scheme considered the stability is proved and a quasi optimal error estimate for the velocity in l l 2 norm is derived

numerical methods for evolutionary reaction diffusion problems with nonlinear reaction terms we have developed new methods to integrate efficiently reaction diffusion parabolic problems with non linear reaction terms in order to obtain uniform and unconditional convergence such methods combine the advantages of alternating direction methods the additive runge kutta methods designed by cooper and sayfy for nonlinear stiff problems as well as the use of shishkin meshes in the singularly perturbed case the resulting algorithms are only linearly implicit and they have the same order of computational complexity per time step that any explicit method we show some numerical experiences which illustrate the good properties of our schemes predicted by the theoretical results

a novel exponentially fitted triangular finite element method for an advection diffusion problem with boundary layers in this paper we develop an exponentially fitted finite element method for a singularly perturbed advection diffusion problem with a singular perturbation parameter this finite element method is based on a set of novel piecewise exponential basis functions constructed on unstructured triangular meshes the basis functions can not be expressed explicitly but the values of each of them and its associated flux at a point are determined by a set of two point boundary value problems which can be solved exactly a method for evaluating elements of the stiffness matrix is also proposed for the case that is small numerical results presented to validate the method show that the method is stable for a large range of it is also shown by the numerical results that the rate of convergence of the method in an energy norm is of orderh1 2when is small

overlapping schwarz methods for a singularly perturbed semilinear elliptic problem and their parallel implementation domain decomposition algorithms based on the schwarz alternating method are applied to the numerical solution of a singularly perturbed semilinear elliptic problem the emphasis is given to the description of the algorithms multidomain schwarz alternating method and two level schwarz method as well as their computer realization on a distributed memory multiprocessor computer numerical experiments related to a model problem illustrate the performance of the algorithms on parallel environment as well as their behavior with respect to the critical parameters such as the perturbation parameter the amount of overlapping and sizes of subdomains application of the two level schwarz method to a model semiconductor problem is also discussed

numerical analysis of singularly perturbed delay differential equations with layer behavior in this paper we present a numerical method to solve boundary value problems for singularly perturbed differential difference equations with negative shift in recent papers the term negative shift has been used for delay siam j appl math 54 1994 249 siam j appl math 42 1982 502 siam j appl math 45 1985 687 siam j appl math 45 1985 708 siam j appl math 54 1994 273 similar bvps are associated with expected first exit time problem of the membrane potential in models for neuron and in variational problem in control theory the stability and convergence analysis of the method is discussed also the effect of small shift on the boundary layer solution in both the cases i e boundary layer on the left side as well as the right side is discussed by considering numerical experiments several numerical examples are solved using the presented method compared the computed result with exact solution and plotted the graphs of the solution of the problems

a finite element method for a singularly perturbed boundary value problem we examine the problem eu a x u b x u f x for 0 0 b x 2 4e 0 a b andf inc 2 0 1 e in 0 1 u 0 andu 1 given using finite elements and a discretized green s function we show that the el mistikawy and werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem i e the nodal errors are bounded bych 2 wherec is independent ofh and e with a natural choice of trial functions uniform first order accuracy is obtained in thel 0 1 norm on choosing piecewise linear trial functions hat functions uniform first order accuracy is obtained in thel 1 0 1 norm

an analysis of a conforming exponentially fitted finite element method for a convection diffusion problem in this paper we present a convergence analysis for a conforming exponentially fitted galerkin finite element method with triangular elements for a linear singularly perturbed convection diffusion problem with a singular perturbation parameter e it is shown that the error for the finite element solution in the energy norm is bounded by o h e1 2 u 2 e 1 2 u 1 if a regular family of triangular meshes is used in the case that a problem contains only exponential boundary layers the method is shown to be convergent at a rate of h1 2 h ln e on anisotropic layer fitted meshes

global convergence method for singularly perturbed boundary value problems we give uniformly convergent splines difference scheme for singularly perturbed boundary value problems eu p x u q x u f x u a 0 u b 1 1 by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration we prove the o min h2 e2 order of uniform convergence with respect to small parameter e at nodes on uniform mesh and o min h e order of uniform global convergence with respect to the approximate solution given by s x i 1n s i x h xi x where h is the heaviside function which is the approximation for the closed form of the exact solution

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

parallel galerkin domain decomposition procedures based on the streamline diffusion method for convection diffusion problems based upon the streamline diffusion method parallel galerkin domain decomposition procedures for convection diffusion problems are given these procedures use implicit method in the sub domains and simple explicit flux calculations on the inter boundaries of sub domains by integral mean method or extrapolation method to predict the inner boundary conditions thus the parallelism can be achieved by these procedures the explicit nature of the flux calculations induces a time step limitation that is necessary to preserve stability artificial diffusion parameters d are given by analysis optimal order error estimate is derived in a norm which is stronger than l 2 norm for these procedures this error estimate not only includes the optimal h 1 norm error estimate but also includes the error estimate along the streamline direction b u u which cannot be achieved by standard finite element method experimental results are presented to confirm theoretical results

optimal b spline collocation method for self adjoint singularly perturbed boundary value problems we present a b spline collocation method of higher order for a class of self adjoint singularly perturbed boundary value problems the essential idea in this method is to divide the domain of the differential equation into three non overlapping subdomains and solve the regular problems obtained by transforming the differential equation with respective boundary conditions on these subdomains using the present higher order b spline collocation method the boundary conditions at the transition points are obtained by the asymptotic approximation of order zero to the solution of the problem the convergence analysis is given and the method is shown to have optimal order convergence by collocating the perturbed differential equation which is satisfied by a special cubic spline interpolate of the true solution numerical experiments are conducted to demonstrate the efficiency of the method

on adaptive vector transform quantization for speech coding adaptive vector transform quantization avtq as a coding system is discussed the optimal bit assignment is derived based on vector quantization asymptotic theory for different pdfs probability density functions of the transform coefficients strategies for shaping the quantization noise spectrum and for adapting the bit assignment to the changes in the speech statistics are discussed a good estimate of the efficiency of any coding system is given by the system coding gain over scalar pcm pulse code modulation based on the optimal bit allocation the coding gain of the vector transform quantization vtq system operating on a stationary input signal is derived the vtq coding gain demonstrates a significant advantage of vector quantization over scalar quantization within the framework of transform coding system simulation results are presented for a first order gauss markov process and for typical speech waveforms the results of fixed and adaptive systems are compared for speech input also the avtq results are compared to known scalar speech coding systems

the solution of singularity perturbed huge problems via domain decomposition abstract this paper deals with iterative algorithms for domain decomposition suitable for singular perturbation problems convergence results are established for one dimensional and two dimensional problems decomposition into subdomains is accompanied via singular perturbation character of problems numerical examples are presented the developed parallel iterative algorithm is compared with the schwarz alternating procedure

image compression by vector quantization of multiresolution decompositions abstract an image compression scheme is introduced which involves a multiresolution decomposition derived from the wavelet transform the use of this transformation in image coding is motivated by its similarity to the processing which occurs at the cortical level of the visual system and by the fact that subband coding in general allows the coder to be better matched to the signal statistics in this work each subband is coded separately by vector quantization this work differs from previous wavelet transform coding schemes involving vector quantization in that the quantization parameters for each subband are selected by a nonlinear optimization procedure which involves constraints on the overall bit rate and the encoding complexity the multiplicative visual model is reviewed and a motivation for cascading this system with the wavelet transform to give a more complete description of human vision is provided empirical results are presented which demostrate the effectiveness of the technique for compressing monochromatic images signal to noise ratio measurements are shown which demostrate the system performance for various selections of bit rate and complexity

a class of singularly perturbed semilinear differential equations with interior layers in this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined a numerical method is constructed for these problems which involves an appropriate piecewise uniform mesh the method is shown to be uniformly convergent with respect to the singular perturbation parameter numerical results are presented that validate the theoretical results

vector predictive coding of speech at 16 kbits s vector quantization in its simplest form may be regarded as a generalization of pcm independent quantization of each sample of a waveform to what might be called vector pcm where a block of consecutive samples a vector is simultaneously quantized as one unit in theory a performance arbitrarily close to the ultimate rate distortion limit is achievable with waveform vector quantization if the dimension of the vector k is large enough the main obstacle in effectively using vector quantization is complexity a vector quantizer of dimension k operating at a rate of r bits sample requires a number of computations on the order of k2 kr and a memory of the same order however a low dimensional vector quantizer dimensions 4 8 achieves a remarkable improvement over scalar quantization pcm consequently using the vector quantizer as a building block and imbedding it with other waveform data compression techniques may lead to the development of a new and powerful class of waveform coding systems this paper proposes and analyzes a waveform coding system adaptive vector predictive coding avpc in which a low dimensionality vector quantizer is used in an adaptive predictive coding scheme in the encoding process a locally generated prediction of the current input vector is subtracted from the current vector and the resulting error vector is coded by a vector quantizer each frame consisting of many vectors is classified into one of m statistical types this classification determines which one of m fixed predictors and of m vector quantizers will be used for encoding the current frame

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

random access to grammar compressed strings and trees grammar based compression where one replaces a long string by a small context free grammar that generates the string is a simple and powerful paradigm that captures sometimes with slight reduction in efficiency many of the popular compression schemes including the lempel ziv family run length encoding byte pair encoding sequitur and re pair in this paper we present a novel grammar representation that allows efficient random access to any character or substring without decompressing the string let s be a string of length n compressed into a context free grammar mathcal s of size n we present two representations of mathcal s achieving o log n random access time and either o n cdot alpha k n construction time and space on the pointer machine model or o n construction time and space on the ram here alpha k n is the inverse of the k th row of ackermann s function our representations also efficiently support decompression of any substring in s we can decompres

compressed range minimum queries abstract given a string s of n integers in 0 a range minimum query rmq i j asks for the index of the smallest integer in s i j it is well known that the problem can be solved with a succinct data structure of size 2 n o n and constant query time in this paper we show how to preprocess s into a compressed representation that allows fast range minimum queries this allows for sublinear size data structures with logarithmic query time the most natural approach is to use string compression and construct a data structure for answering range minimum queries directly on the compressed string we investigate this approach in the context of grammar compression we then consider an alternative approach instead of compressing s using string compression we compress the cartesian tree of s using tree compression we show that this approach can be exponentially better than the former is never worse by more than an o factor i e for constant alphabets it is never asymptotically worse and can in fact be worse by an factor

on a domain decomposition algorithm for a singularly perturbed reaction diffusion problem this paper deals with an iterative algorithm for domain decomposition applied to the solution of a singularly perturbed reaction diffusion problem this algorithm is based on finite difference domain decomposition approach and suitable for parallel computing convergence properties of the algorithm are established numerical results for a test singularly perturbed problem are presented

parallel galerkin domain decomposition procedures for wave equation parallel galerkin domain decomposition procedures for wave equation are given these procedures use implicit method in the sub domains and simple explicit flux calculation on the inter boundaries of sub domains by integral mean method or extrapolation method thus the parallelism can be achieved by these procedures the explicit nature of the flux prediction induces a time step constraint that is necessary to preserve the stability l 2 norm error estimates are derived for these procedures experimental results are presented to confirm the theoretical results

a uniformly convergent finite difference scheme for a singularly perturbed semilinear equation boundary value problems for singularly perturbed semilinear elliptic equations are considered special piecewise uniform meshes are constructed which yield accurate numerical solutions irrespective of the value of the small parameter numerical methods composed of standard monotone finite difference operators and these piecewise uniform meshes are shown theoretically to be uniformly with respect to the singular perturbation parameter convergent numerical results are also presented which indicate that in practice the method is first order accurate

approximation of smallest linear tree grammar a simple linear time algorithm for constructing a linear context free tree grammar of size o r g r g log n r g for a given input tree t of size n is presented where g is the size of a minimal linear context free tree grammar for t and r is the maximal rank of symbols in t which is a constant in many applications this is the first example of a grammar based tree compression algorithm with a good i e logarithmic in terms of the size of the input tree approximation ratio the analysis of the algorithm uses an extension of the recompression technique from strings to trees

two level parallel schwarz methods for singularly perturbed semilinear elliptic problems abstract two level domain decomposition methods based on the schwarz alternating procedure are proposed for the numerical solution of singularly perturbed semilinear elliptic problems convergence properties of the two level schwarz methods are established numerical experiments for a test problem illustrate the behaviour of the methods with respect to the critical parameters such as the perturbation parameter the amount of overlapping and sizes of subdomains application of one of the two level schwarz methods to a model semiconductor problem is also discussed

a class of singularly perturbed quasilinear differential equations with interior layers a class of singularly perturbed quasilinear differential equations with discontinuous data is examined in general interior layers will appear in the solutions of problems from this class a numerical method is constructed for this problem class which involves an appropriate piecewise uniform mesh the method is shown to be a parameter uniform numerical method with respect to the singular perturbation parameter numerical results are presented which support the theoretical results

finite difference domain decomposition algorithms for a parabolic problem with boundary layers abstract parallel algorithms combining a time discretization and finite difference domain decomposition methods are applied to solution of a singularly perturbed parabolic problem different time discretizations are applied in different subdomains convergence results for the algorithms are established

explicit implicit and crank nicolson domain decomposition methods for parabolic partial differential equations abstract two parallel non overlapping domain decomposition algorithms for solving parabolic partial differential equations are proposed the algorithms combine crank nicolson scheme with implicit galerkin finite element methods in sub domains and explicit flux approximation along inner boundaries at each time step thus parallelism can be easily achieved l 2 norm error estimates for these explicit implicit procedures are presented in which time step constraints are proved to be less severe than that of fully explicit schemes numerical experiments are also performed to verify the theoretical analysis

comparison of parallel implementation of some multi level schwarz methods for singularly perturbed parabolic problems abstract parallel multi level algorithms combining a time discretization and an overlapping domain decomposition technique are applied to the numerical solution of singularly perturbed parabolic problems two methods based on the schwarz alternating procedure are considered a two level method with auxiliary correcting subproblems as well as a three level method with auxiliary predicting and correcting subproblems moreover modifications of the methods using time extrapolation on subdomain interfaces are investigated the emphasis is given to the description of the algorithms as well as their computer realization on a distributed memory multiprocessor computer numerical experiments illustrate the performance of the algorithms on parallel environment and their behaviour with respect to the critical parameters such as the perturbation parameter and the size of the auxiliary subdomains

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

inadequacy of first order upwind difference schemes for some recirculating flows spurious numerical solutions of problems with closed sub characteristics by upwind difference schemes in particular problems of recirculating incompressible flow at high reynolds numbers are proved to be due to the anisotropy of the artificial viscosity numerical examples are presented to show that even very simple problems including basic problems in fluid dynamics are not approximated well by schemes with anisotropic artificial viscosity regardless of numerical parameters 1991 academic press inc numerical methods for solving incompressible fluid flow equations are frequently tested on problems of recirculating flow quite often results obtained by different methods differ from each other significantly reliable experimental data is usually difficult to obtain and discrepancies between numerical two dimensional solutions and experimental data may be the result of the influence of walls in the experiments the problems in solving recirculating incompressible flows at high reynolds numbers are well known several researchers have pointed out that some of the methods employed to obtain stable discretization e g upwind differencing may lead to spurious results such results have been attributed to excessive artificial viscosity in the numerical scheme to multiple solutions of the non linear set of algebraic equations that is obtained from the numerical scheme and to poor resolution by grids that are too coarse 2 4 51 but spurious results occur even when none of these reasons are valid indeed a single linear partial differential equation may exhibit such behavior when the sub characteristics are closed this is true even when the mesh size is small enough to easily resolve the solution and truncation errors are genuinely small everywhere and even when the differential solution does not depend on the

a nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection diffusion equation in this paper we formulate and analyze a discretization method for a 2d linear singularly perturbed convection diffusion problem with a singular perturbation parameter e the method is based on a nonconforming combination of the conventional galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method and on a mixture of triangular and rectangular elements it is shown that the method is stable with respect to a semi discrete energy norm and the approximation error in the semi discrete energy norm is bounded by ch ln e ln h with c independent of the mesh parameter h the diffusion coefficient e and the exact solution of the problem

solution of a singularly perturbed parabolic problem by domain decomposition with time extrapolation parallel iterative algorithms combining a time discretization and domain decomposition methods are applied to solution of a singularly perturbed parabolic problem the domain decomposition methods are based on the schwarz alternating procedure and time extrapolation convergence results for the iterative algorithms are established numerical experiments are presented to compare the effectiveness of the iterative algorithms on a shared memory multiprocessor

optimum design of vector quantized subband codecs this article provides an approach for representing an optimum vector quantizer by a scalar nonlinear gain plus additive noise model the validity and accuracy of this analytic model is confirmed by comparing the calculated model quantization errors with actual simulation of the optimum linde buzo gray 1980 vector quantizer using this model we form an mse measure of an m band filter bank codec in terms of the equivalent scalar quantization model and find the optimum fir filter coefficients for each channel in the m band structure for a given bit rate filter length and input signal correlation model specific design examples are worked out for four tap filters in the two band paraunitary case these theoretical results are confirmed by extensive monte carlo simulation

the finite element method with weighted basis functions for singularly perturbed convection diffusion problems in this paper we present a finite element method for singularly perturbed convection diffusion problems in both one and two dimensions based on a set of weighted basis functions constructed on unstructured meshes in 2d for the one dimensional case both first and second order schemes are discussed a technique for approximating fluxes is proposed some theoretical results on uniform convergence are obtained for the two dimensional case a first order scheme is constructed for problems with two singular perturbation parameters a technique is also developed in approximating fluxes in 2d this technique is used to simplify the calculation of the integrals in the stiffness matrix arising from the scheme which will save computational costs the numerical results support the theoretical results and demonstrate that the method is stable for a wide range of singular perturbation parameters

singularly perturbed problems in partial differential equations a survey this survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the methods developed by numerous researchers after 1980 2000 in this review we have covered singularly perturbed partial differential equations about odes the survey has already been done by us see m k kadalbajoo k c patidar appl math comput 130 2002 457 510

a streamline diffusion method for nonconforming finite element approximations applied to convection diffusion problems abstract we consider a nonconforming streamline diffusion finite element method for solving convection diffusion problems the theoretical and numerical investigation for triangular and tetrahedral meshes recently given by john maubach and tobiska has shown that the usual application of the sdfem gives not a sufficient stabilization additional parameter dependent jump terms have been proposed which preserve the same order of convergence as in the conforming case the error analysis has been essentially based on the existence of a conforming finite element subspace of the nonconforming space thus the analysis can be applied for example to the crouzeix raviart element but not to the nonconforming quadrilateral elements proposed by rannacher and turek in this paper parameter free new jump terms are developed which allow to handle both the triangular and the quadrilateral case numerical experiments support the theoretical predictions

an exponentially fitted finite volume method for the numerical solution of 2d unsteady incompressible flow problems abstract in this paper we develop and test an exponentially fitted finite volume method for the numerical solution of the navier stokes equations describing 2d incompressible flows the method is based on an unstructured delaunay mesh and its dual dirichlet tessellation combined with a locally constant approximation to the flux this yields a piecewise exponential approximation to the exact solution numerical tests are presented for a linear advection diffusion problem with boundary layers the method is then applied to the driven cavity problem with reynolds numbers up to 10 4 the numerical results indicate that the method is robust for a wide range of values of the reynolds number in the case re 10 4 unsteady solutions are captured if the mesh is sufficiently fine

domain decomposition procedures combined with h1 galerkin mixed finite element method for parabolic equation abstract non overlapping domain decomposition procedures are considered for parabolic equation these procedures are combined with using h 1 galerkin mixed finite element method in the sub domains to approximate the primary variable u and its flux simultaneously explicit calculations are built by using integral mean methods to present the inter domain boundary conditions for the flux thus the parallelism can be achieved by these procedures two approximation schemes are established time step constraints are proved necessary to preserve stability which are less severe than that of fully explicit galerkin finite element method the mixed finite element spaces are allowed to be of different polynomial degrees and not subject to the lbb consistency condition new nonstandard elliptic projections are defined and analyzed optimal error estimates for the variable u in h 1 norm and its flux in l 2 norm and are derived for these schemes numerical experiments are presented to confirm the theoretical results

iterative domain decomposition algorithms for the solution of singularly perturbed parabolic problems abstract a combination of a time discretization and domain decomposition methods for the solution of singularly perturbed semilinear parabolic problems is considered on this basis iterative algorithms suitable for parallelization are constructed convergence properties of these algorithms are established the implementation of the algorithms on a shared memory multiprocessor is examined numerical results for a test singularly perturbed problem are presented

computational method for a singular perturbation problem via domain decomposition and its parallel implementation abstract in this paper a computation method for the nonlinear problem with a small perturbation parameter via domain decomposition is presented we apply a combination of the two approaches to solving the problem iterative algorithms for domain decomposition and a grid refinement technique on subdomains two iterative algorithms are examined the first one is the schwarz alternating procedure and the second algorithm is highly suitable for parallel computing numerical examples are provided

joint bit allocation and dimensions optimization for vector transform quantization in vector transform quantization vtq vectors consisting of m consecutive samples of a waveform are transformed into a set of m coefficients that are quantized by m

multiplicative schwarz methods for parabolic problems the class of multiplicative schwarz methods originated from the classical schwarz alternating method it has been shown to be one of the most powerful methods for solving finite element or finite d

a tetrahedral mixed finite element method for the stationary semiconductor continuity equations a petrov galerkin mixed finite element method based on tetrahedral elements for the stationary semiconductor device continuity equations is presented this method can be regarded as a natural extension to three dimensions of the well known scharfetter gummel one dimensional scheme existence uniqueness and stability of the approximate solution are proved and an error estimate is given for an arbitrary delaunay mesh and corresponding dirichlet tessellation the associated linear system is a symmetric and positive definite m matrix the evaluation of the terminal currents associated with the method is also discussed and it is shown that the computed terminal currents are convergent and conservative numerical results for a model three dimensional device are presented to validate the theoretical results

a solution of the discrepancy occurs due to using the fitted mesh approach rather than to the fitted operator for solving singularly perturbed differential equations abstract the solution of the boundary value problems for singularly perturbed differential equations i e where the highest order derivative is multiplied by a small parameter exhibits layer behavior the classical numerical schemes to solve such types of the boundary value problems do not give satisfactory result when the perturbation parameter is sufficiently small to resolve this difficulty there are mainly two approaches namely fitted operator and fitted mesh both the scheme are uniformly convergent i e their convergence is independent of the small perturbation parameter it is justified to adopt the two approach rather than the classical numerical schemes to solve the boundary value problems for singularly perturbed differential equations now if we compare the two approaches one thing is common that both the approaches give the parameter uniform schemes which is the primary requirement in construction of the numerical scheme to solve such type of problem secondly one desire a higher order numerical scheme to approximate the solution of a problem as far as order of convergence is concerned the numerical scheme based on fitted operator approach is better than the numerical scheme constructed using fitted mesh approach the researchers who adopted the fitted mesh rather than fitted operator approach to solve a singularly perturbed problem faced the question due to the loss of order of convergence most of them justified it by quoting the simplicity of the method and there are some non linear problems for which a parameter uniform scheme cannot be constructed based on fitted operator approach while for the same problem a parameter uniform scheme is constructed based on fitted mesh method now question remains unanswered in the case of linear problem in this article we replied to this question by giving an example of linear problem for which one cannot construct a parameter uniform scheme based on fitted operator approach while for the same problem a parameter uniform numerical scheme based fitted mesh approach has been constructed m k kadalbajoo k k sharma e uniform fitted mesh method for singularly perturbed differential difference equations with mixed type of shifts with layer behavior int j comput math 81 2004 49 62 a theoretical reason behind the inability in construction of a parameter uniform scheme using fitted operator approach is revealed in support of the predicted theory a number of numerical experiments are carried out

an optimal design procedure for intraband vector quantized subband coding subband coding with vector quantization is addressed in this paper forming the data vectors from both between and within the subbands is considered the former of these two schemes is referred to as interband coding and the latter as intraband coding interband coder design is relatively straightforward since the design of the single codebook involved follows readily from a representative set of interband data vectors intraband coder design is more complicated since it entails the selection of a vector dimension and a bit rate for each subband the main contribution of this work is an optimal methodology for intraband subband vector quantizer design the problem formulation includes constraints on the bit rate and the encoding complexity and is solved with nonlinear programming methods subband vector quantization image coding in conjunction with a human visual system model is thoroughly investigated results of a large number of experiments indicate that the optimal intraband coder yields superior results from quantitative as well as subjective points of view than the interband coder for comparable bit rates this improvement becomes more pronounced as the computational complexity of the intraband encoder is allowed to increase

lrm trees compressed indices adaptive sorting and compressed permutations lrm trees are an elegant way to partition a sequence of values into sorted consecutive blocks they have been used to encode ordinal trees and to index integer arrays in order to support range minimum queries on them we describe how they yield many other convenient results in a variety of areas compressed indices for range minimum queries on partially sorted arrays a new adaptive sorting algorithm and a compressed data structure for permutations supporting direct and inverse application in time inversely proportional to the compressibility of the permutation

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100