homotopical stable ranks for banach algebras abstract the connected stable rank and the general stable rank are homotopy invariants for banach algebras whereas the bass stable rank and the topological stable rank should be thought of as dimensional invariants this paper studies the two homotopical stable ranks viz their general properties as well as specific examples and computations the picture that emerges is that of a strong affinity between the homotopical stable ranks and a marked contrast with the dimensional ones

banach valued holomorphic functions on the maximal ideal space of h we study banach valued holomorphic functions defined on open subsets of the maximal ideal space of the banach algebra h of bounded holomorphic functions on the unit disk mathbb d subset mathbb c with pointwise multiplication and supremum norm in particular we establish vanishing cohomology for sheaves of germs of such functions and solving a banach valued corona problem for h prove that the maximal ideal space of the algebra h mathrm comp infty a of holomorphic functions on mathbb d with relatively compact images in a commutative unital complex banach algebra a is homeomorphic to the direct product of maximal ideal spaces of h and a

algebras of almost periodic functions with bohr fourier spectrum in a semigroup hermite property and its applications abstract it is proved that the unital banach algebra of almost periodic functions of several variables with bohr fourier spectrum in a given additive semigroup is an hermite ring the same property holds for the wiener algebra of functions that in addition have absolutely convergent bohr fourier series as applications of the hermite property of these algebras we study factorizations of wiener hopf type of rectangular matrix functions and the toeplitz corona problem in the context of almost periodic functions of several variables

reliable and robust a posteriori error estimation for singularly perturbed reaction diffusion problems problems with singular perturbations exhibit solutions with strong boundary layers and other types of local behavior such features lend themselves to adaptive solution methods the quality of any adaptive algorithm ultimately rests on the reliability and robustness of the a posteriori error control an estimator that has proved to be one of the most reliable is the equilibrated residual method the main property of the estimator is that it bounds the true error from above however the method is not robust in the singularly perturbed limit the current work generalizes the error estimator based on the equilibrated residuals and coincides with the standard method in the unperturbed limit it is shown that the new method is robust in the singularly perturbed limit while maintaining reliability yielding a guaranteed upper bound on the true error finally the application of the estimator to the problem of controlling the spatial error in rothe s method for the time discretization of a simple parabolic problem is included

n widths and singularly perturbed boundary value problems using the theory of n widths the approximability of solutions of singularly perturbed linear reaction diffusion problems in two dimensions is quantified

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 profile of solutions with two sharp layers to a singularly perturbed semilinear dirichlet problem we discuss the existence of positive solutions of some singularity perturbed elliptic equations on convex domains with nonlinearity changing sign in particular we obtain solutions with both a boundary layer and a sharp interior peak

experiments with a local adaptive grid h refinement for the finite difference solution of bvps in singularly perturbed second order odes abstract despite many years of the development of adaptive grid methods there is still a need for re evaluation tuning and comparisons of the various approaches to grid adaptation in this study we seek to identify the most reliable and computationally inexpensive variant of the iterative grid refinement de refinement algorithm for the finite difference solution of singularly perturbed bvps in second order odes that is also easily extendable to evolutionary pdes conventional three point discretisations of the first and second derivatives are used errors of the discrete solution and its first derivative are simultaneously controlled the grid refinement is accomplished by adding nodes at inter nodal positions and de refinement by removing previously added nodes in reverse order computational experiments are performed for 15 examples of odes assuming several alternative choices of the a posteriori error estimators grid refinement indicators and regridding strategies the most satisfactory results are obtained by combining deferred approach to error estimation based on higher order reference derivative approximations for truncation error evaluation interpolation based indicator for grid refinement and regridding strategy that uses the mean indicator value as the indicator threshold for refinement

sufficient conditions for the projective freeness of banach algebras let r be a unital semi simple commutative complex banach algebra and let m r denote its maximal ideal space equipped with the gelfand topology sufficient topological conditions are given on m r for r to be a projective free ring that is a ring in which every finitely generated projective r module is free several examples are included notably the hardy algebra h x of bounded holomorphic functions on a riemann surface of finite type and also some algebras of stable transfer functions arising in control theory

power series for inverse jacobian elliptic functions the 12 inverse jacobian elliptic functions are expanded in power series by using properties of the symmetric elliptic integral of the first kind suitable notation allows three series to include all 12 cases three of which have been given previously all coefficients are polynomials in the modulus k that are homogeneous variants of legendre polynomials the four series in each of three subsets have the same coefficients in terms of k

the solution of singularly perturbed semilinear elliptic problems by iterative domain decomposition algorithms abstract parallel domain decomposition algorithms for the solution of singularly perturbed semilinear elliptic problems are considered convergence properties and numerical stability of these iterative algorithms are established numerical experiments are provided

an adaptive finite element method for unsteady convection dominated flows with stiff source terms we consider nonstationary convection dominated flows with stiff source terms as a unified approach to such problems a combined finite element method in space and time with streamline diffusion is examined numerically it has good shock capturing features and is implicit and l stable moreover being a galerkin method it admits residual based weighted a posteriori error estimates of optimal order we avoid the use of global stability constants by actually solving the dual problem this in turn leads to efficient and mathematically rigorous mesh refinement strategies where the streamline diffusion parameter is used for optimizing the resulting adaptive scheme all theoretical results are substantiated by numerical test examples for a physical application we turn to detonation waves at moderate relaxation times in one space dimension the finite element method well reproduces the chapman jouguet speed of detonations and their wave structure our error estimator proves accurate and sensitive to parameters at least on time intervals of moderate length the adaptive scheme based on this estimator produces fairly accurate results which appear to be the best available with the present strategy all ingredients of the numerical scheme can be extended in a natural way to higher space dimensions

direct expansion method of boundary condition for solving 3d elliptic equations with small parameters in the irregular domain in this article a new methodology direct expansion method of boundary condition dembc is developed to solve 3d elliptic equations in the irregular domain first the previous rational differential quadrature method rational spectral collocation method in berrut et al 2005 8 developed by berrut et al 2005 8 has been generalized to solve 3d elliptic equations second it is showed that direct expansion method of boundary condition is capable of handling boundary problems with higher efficiency finally with the help of conformal mapping tee and trefethen 2006 9 and domain decomposition method dembc and 3d rdqm are able solve three kinds of 3d elliptic equations with small parameters in the irregular domain numerous test results justify the accuracy and efficiency of our approach

on some singularly perturbed problems for viscous stratified fluids abstract the singularly perturbed boundary value problem which describes the motion of a stratified viscous fluid is obtained and an asymptotic expansion of the solution for vanishing viscosity is developed the two cases considered are when the reduced equation has i a unique solution and ii a family of solutions the latter situation is an example of a singular singular perturbation problem proof of the uniform asymptotic validity of the constructed expansion is given

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

gradient estimates for variable coefficient parabolic equations and singular perturbation problems in this article we prove via monotonicity formulas interior and boundary gradient esti mates for solutions to second order parabolic equations in divergence form with dini top order coefficients we then prove uniform lipschitz estimates for solutions of singular perturbation prob lems using the previous results and two phase monotonicity formulas introduction the purpose of this paper is to study the regularity of spatial gradients of solutions to second order uniformly parabolic equations in divergence form with bounded lower order terms and dini continuous coefficients and also to establish uniform spatial lipschitz estimates for some singular perturbation problems in the same context in part 1 we obtain interior and boundary estimates for solutions to these linear equations as is well known the weakest assumption on the modulus of continuity of the top order terms to obtain interior continuity of the gradient of solutions is dini continuity see part 1 for the relevant definitions see for instance g w in the elliptic case the usual approach to such results is to use appropriate potentials here we use a monotonicity formula approach theo rem 1 2 9 which we believe is new the approach has the advantage that it allows us to show that dini continuity of the coefficients at a given point yields bound edness of the gradient at the same point using our monotonicity formula the fact that linear functions are almost solutions and a geometric argument from cl we also obtain under uniform dini continuity of the coefficients a uniform modulus of continuity estimate for gradients of solutions theorem 1 3 1 the fact that the monotonicity formula at a point depends only on the dini continuity at that same point enables us to use a simple reflection argument to extend these results up to the boundary theorem 1 4 5 and theorem 1 4 12 we thus obtain a modulus of continuity estimate up to the boundary for gradients of solutions with smooth dirichlet data under uniform dini continuity of the top order coeffi cients once this is established a compactness argument theorem 1 5 10 and a judicious application of harnack s principle allow us to obtain a hopf maximum principle theorem 1 5 10 corollary 1 5 13 and corollary 1 5 16 for our class of equations

some results on linear rational trigonometric interpolation abstract we present here formulae for calculating the p th derivative of a linear rational trigonometric interpolant written in barycentric form we give sets of interpolating points for which the interpolant converges exponentially towards the interpolated function

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

grauert and lax halmos type theorems and extension of matrices with entries in h abstract in the paper we prove an extension theorem for matrices with entries in h u for u a riemann surface of a special type one of the main components of the proof is a grauert type theorem for holomorphic vector bundles defined on maximal ideal spaces of certain banach algebras

e uniformly convergent fitted mesh finite difference methods for general singular perturbation problems abstract we consider general singular perturbation problems of the form cey x a x y x b x y x f x x 0 1 y 0 0 y 1 1 with ce equals to both e and e a x b x f x are positive throughout the interval and 0 1 r the indirect methods those which do not use any acceleration of convergence techniques e g richardson s extrapolation or defect correction etc for such problems on a mesh of shishkin type lead the error as o n 1 ln n where n denotes the total number of sub intervals of 0 1 in this paper we systematically describe a very simple and direct method which reduces the error to o n 2 ln 2 n this method is proved to be e uniformly convergent with the above error bounds on a piecewise uniform mesh of shishkin type the motivation for using this shishkin mesh is inspired by the quotation of stynes m stynes a jejune heuristic mesh theorem comput methods appl math 3 2003 488 492 miller has moved from j j h miller construction of a fem for a singularly perturbed problem in 2 dimensions in numerische behandlung von differentialgleichungen band 2 tagung math forschungsinst oberwolfach 1975 internat ser numer math birkhauser basel vol 31 1976 pp 165 169 the question what scheme should one use on a given mesh to p a farrell a f hegarty j j h miller e o riordan g i shishkin robust computational techniques for boundary layers chapman hall crc new york 2000 what mesh should one use with a given scheme the theoretical estimates have been justified by several numerical examples

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

stability analysis of coupled linear ode hyperbolic pde systems with two time scales abstract this paper is concerned with a class of coupled ode pde systems with two time scales the fast constant time scale is modeled by a small positive perturbation parameter first we state a general sufficient stability condition for such systems this condition is also sufficient for the stability of the reduced and boundary layer subsystems however counterexamples illustrate that the converse is not true next we study the stability of such systems by taking into account the fact that the perturbation parameter is sufficiently small for linear ode coupled with fast hyperbolic pde systems the stability of both subsystems implies the stability of the full system on the other hand a counterexample shows that the full system can be unstable even though the two subsystems are stable for a pde coupled with fast ode system numerical simulations on academic examples are proposed moreover an application to boundary control of a gas flow transport system is used to illustrate the theoretical result

cech cohomology and covering dimension for the h maximal ideal space it is proved that the n cech cohomology groups of the h maximal ideal space x h are trivial for n 2 and that single elements of h separate points from closed sets in x h these results are used to prove that the covering dimension of x h is 2

high order fitted operator numerical method for self adjoint singular perturbation problems we consider self adjoint singularly perturbed two point boundary value problems in conservation form highest possible order of uniform convergence for such problems achieved hitherto via fitted operator methods was one see e g doolan et al uniform numerical methods for problems with initial and boundary layers boole press dublin 1980 p 121 reducing the original problem into the normal form and then using the theory of inverse monotone matrices a fitted operator finite difference method is derived via the standard numerov s method the scheme thus derived is fourth order accurate for moderate values of the perturbation parameter e whereas for very small values of this parameter the method is e uniformly convergent with order two numerical examples are given in support of the theory

uniformly convergent difference schemes for singularly perturbed parabolic diffusion convection problems without turning points we consider first the initial boundary value problem for the parabolic equation varepsilon y xx a x t y x b x t y d x t y t f x t on 0 1 0 t where 0 0 b 0 andd 0 bounds ony and its derivatives are obtained under certain compatibility assumptions on the data we examine a family of difference schemes for this problem which is exponentially fitted in thex variable and uses classical differencing in thet variable on rectangular grids which are arbitrarily spaced in both thex andt directions under a courant friedrichs lewy type condition the errors at the grid points are shown to be bounded byc h k whereh k is the maximum mesh width in thex t direction andc is a constant independent of h andk the corresponding result for a two point boundary value problem is also derived

the hyperbolic singular perturbation problem an operator theoretic approach abstract we study the convergence of solutions of e2u t u t e2a b u t to solutions of u t bu t as e 0 where a and b are linear unbounded operators in a banach space

on solutions of some unsteady flows over a continuous moving porous flat surface abstract a singular perturbation solution for hydromagnetic boundary layer flow over a continuously moving surface with large suction is obtained and compared with its exact solution also the solutions are obtained for finite and large values of t time and for various asymptotic cases flow of this type represents a new class of boundary layer problems with solutions substantially different from those for boundary layer flow on a flat surface of finite length surprisingly the inner solution of the problem is valid even in the outer region and the effect of magnetic field on the flow field is almost negligible furthermore it is shown that a straightforward perturbation expansion valid for all t time and y space coordinate for small suction is not possible because of the infinite condition on y

necessary l2 uniform convergence conditions for difference schemes for two dimensional convection diffusion problems we derive necessary conditions for the uniform convergence with respect to the perturbation parameter in l2 of schemes for singularly perturbed convection diffusion problems applying these conditions we show that the convergence order of the streamline diffusion finite element method with piecewise linear trial functions cannot be better than 12 uniformly in the singular perturbation parameter

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

topology of the maximal ideal space of h revisited abstract let m h be the maximal ideal space of the banach algebra h of bounded holomorphic functions on the unit disk d c we prove that m h is homeomorphic to the freudenthal compactification m a of the set m a of all non trivial analytic disks gleason parts of m h also we give alternative proofs of important results of suarez asserting that the set m s of trivial one pointed gleason parts of m h is totally disconnected and that the cech cohomology group h 2 m h z 0

artificial boundary conditions for the linear advection diffusion equation a family of artificial boundary conditions for the linear advection diffusion equation with small viscosity is developed well posedness for the associated initial boundary value problem is analyzed the error produced by truncating the domain is estimated numerical results are presented 1 introduction when computing the solution of a partial differential equation in an unbounded domain one often introduces artificial boundaries in order to limit the computational cost these boundaries must be chosen not too far from the domain of interest therefore the boundary conditions must be good approxima tions to the so called transparent i boundary condition i e such ithat the solution of the problem in the bounded domain is equal to the solution in the original domain the transparent boundary condition is usually an integral relation in time and space between u and its normal derivative on the boundary which makes it impractical from a numerical point of view one must approximate this relation to get local boundary conditions they are often called absorbing or artificial boundary conditions this question is of crucial interest in such different areas as geophysics plasma physics fluid dynamics 1 2 3 and the use of such conditions is now classical in geophysics our interest for the linear advection diffusion equation comes from the navier stokes equation but it arises also in other fields as for example meteorology 6 the incompressible navier stokes equation can be written as

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

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

high order finite element methods for singularly perturbed elliptic and parabolic problems we develop a framework for applying high order finite element methods to singularly perturbed elliptic and parabolic differential systems that utilizes special quadrature rules to confine spurious effects such as excess diffusion and nonphysical oscillations to boundary and interior layers this approach is more suited for use with adaptive mesh refinement and order variation techniques than other problem dependent methods quadrature rules developed for two point convection diffusion and reaction diffusion problems are used with finite element software to solve examples involving ordinary and partial differential equations numerical artifacts are confined to layers for all combinations of meshes orders and singular perturbation parameters that were tested radau or lobatto quadrature used with the finite element method to solve respectively convection and reaction diffusion problems provide the benefits of the specialized quadrature formulas and are simpler to implement

robust convergence of multilevel algorithms for convection diffusion equations a new approach is developed to analyze convergence of multilevel algorithms for convection diffusion equations this approach uses a multilevel recursion formula which can be applied to a variety of nonsymmetric problems here the recursion formula is applied to a robust multilevel algorithm for convection diffusion equations with convection in the x or y direction the multilevel algorithm uses semicoarsening line relaxation and prewavelets the convergence rate is proved to be less than 0 18 independent of the size of the convection term and the number of unknowns the assumptions allow the convection term to have a turning point so that an interior layer can appear in the solution of the convection diffusion equation the computational cost of the multilevel cycle is about o n log n independent of the size of the convection term where n is the number of unknowns it is proved that o log n multilevel cycles starting from the initial guess 0 lead to an o n 2 algebraic error with respect to the l infty norm independent of the size of the convection term

a comparison of uniformly convergent difference schemes for two dimensional convection diffusion problems galerkin and petrov galerkin finite element methods are used to obtain new finite difference schemes for the solution of linear two dimensional convection diffusion problems numerical estimates are made of the rates of convergence of these schemes uniformly with respect to the perturbation parameter and these uniform rates are shown to compare favourably with those of established methods

singular perturbation margin and generalised gain margin for linear time invariant systems in this paper we propose a singular perturbation margin spm and a generalised gain margin ggm as stability metrics for single input single output siso linear time invariant lti systems from the view of singular perturbations and regular perturbations which have bijective correspondences with the classical phase margin pm and the gain margin gm respectively both of the numerical and analytical time domain spm and ggm assessment methods are provided and relationships between the singular perturbation parameter pm of the perturbed system pm and spm of the nominal system and the monotonically increasing phase of the fast system are also revealed these results make it possible to assess the pm of the nominal system in the time domain for siso lti systems using the spm with a standardised testing system called pm gauge as demonstrated by examples the concepts of spm and ggm can be used as metrics of stability margins for linear time varying systems and nonlinear systems

calculation of the taylor series expansion coefficients of the jacobian elliptic function introduction not much is known about the taylor series expansion coefficients of the jacobian elliptic functions sn x k cn x k and dn x k in handbooks only the first four or five terms are given see for instance abramowitz and stegun 1 p 575 hancock s book on elliptic functions 4 p 252 and p 486 or gradshteyn and ryshik 5 recently however alois schett gave a combinatorial expression of the coefficients of the taylor series expansion of sn x k and evaluated explicitly the first eight nontrivial terms see 6 p 146 his results were extended by dominique dumont 3 who gave a new combinatorial interpretation of the coefficients of sn x k it is often said that no recurrence formulae exist for these coefficients 1 p 575 or 6 p 143 what is meant by that is probably that no simple recurrence formulae exist using differential and other equations one can deduce a number of useful recurrence relations and with the help of the burmann lagrange theorem even a definite although hard to handle expression of the coefficients in question

uniform convergence of the schwarz alternating method for solving singularly perturbed advection diffusion equations our purpose in this paper is to study the convergence rate of some variants of the schwarz alternating method for solving two types of singularly perturbed advection diffusion equations one in which the advection term first order term dominates and the other in which the lowest order term zeroth order term dominates for such problems boundary or other layer regions may be present for singular perturbation problems in which the advection dominates we choose two overlapping subregions one corresponding to an outer or elliptic region and the other corresponding to an inner or hyperbolic region our main result for this problem shows that if the subdomains can be chosen to follow the flow i e if the boundary interface of one of the subdomains corresponds to an outflow boundary for the streamlines of the flow then the schwarz iterates converge in the maximum norm with an error reduction factor per iteration that exponentially decays with increasing overlap or decreasing diffusion for the singular perturbation problem in which the lowest order term dominates we consider many overlapping subregions with no special requirement on outflow or inflow boundaries our main result here which is valid for two variants of the classical schwarz algorithm shows that the error reduction factor per iteration decays exponentially in the maximum norm with increasing overlap and increasing dominance of the zeroth order term for both problems our proofs rely on the maximum principle and the construction of barrier functions

singular perturbation approximation by means of a h 2 lyapunov function for linear hyperbolic systems abstract a linear hyperbolic system of two conservation laws with two time scales is considered in this paper the fast time scale is modeled by a small perturbation parameter by formally setting the perturbation parameter to zero the full system is decomposed into two subsystems the reduced subsystem representing the slow dynamics and the boundary layer subsystem standing for the fast dynamics the solution of the full system can be approximated by the solution of the reduced subsystem this result is obtained by using a h 2 lyapunov function the estimate of the errors is the order of the perturbation parameter for all initial conditions belonging to h 2 and satisfying suitable compatibility conditions moreover for a particular subset of initial conditions more precise estimates are obtained the main result is illustrated by means of numerical simulations

a global uniformly convergent finite element method for a quasi linear singularly perturbed elliptic problem abstract in this paper we construct a bilinear finite element method based on a special piecewise uniform mesh for solving a quasi linear singularly perturbed elliptic problem in two space dimensions a quasi optimal global uniform convergence rate o n 2x ln2 nx n 2y ln2 ny was obtained which is independent of the perturbation parameter here nx and ny are the number of elements in the x and y directions respectively

a singularly perturbed model problem for numerical computation abstract in this note we introduce a model problem for the numerical solution of a two dimensional singular perturbation problem to combine a number of typical difficulties in a relatively simple problem we propose to solve the linear convection diffusion problem in the domain exterior of a circle we describe the analytical solution of the problem and we comment on its numerical evaluation for small values of the parameter asymptotic approximations of the solution are given based on expansions by friedlander 1958 and waechter 1968 this information gives insight into the behaviour of the solution and allows the computation of a reference solution for small values of the parameter

tikhonov theorem for linear hyperbolic systems abstract a class of linear hyperbolic systems of conservation laws with multiple time scales which are modeled by a perturbation parameter is considered in this paper by setting the perturbation parameter to zero two subsystems the reduced subsystem standing for the slow dynamics and the boundary layer subsystem representing the fast dynamics are computed it is first proved that the exponential stability of the full system implies the stability of both subsystems secondly a counterexample is given to indicate that the stability of the two subsystems does not ensure the full system s stability moreover a new tikhonov theorem for this class of infinite dimensional systems is stated the solution of the full system can be approximated by that of the reduced subsystem and this is proved by lyapunov techniques an application to boundary feedback stabilization of gas transport model is used to illustrate the results

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

a singular perturbation approach to kramers diffusion problem we consider kramers diffusion problem which seeks to calculate the rate of escape of a particle from one potential well over a barrier to another presumably deeper and therefore more stable well though kramers introduced the problem as a model for chemical reactions it applies to numerous rate processes including atomic migration and ionic conductivity in crystals and transitions due to noise between stable states of dynamical systems with multi stable states to name but a few we propose a new approach not based on a fokker planck equation but rather on the solution of a singularly perturbed boundary value problem specifically we relate the rate of escape to the first passage time from the domain of attraction of the stable point corresponding to the first well the first passage time is then characterized via the ito calculus as a solution of an elliptic partial differential equation of singular perturbation type finally this equation is solved asymptotically by methods previously develop

a priori error analysis of residual free bubbles for advection diffusion problems we develop an a priori error analysis of a finite element approximation to the elliptic advection diffusion equation eps delta u conv cdot nabla u f subject to a homogeneous dirichlet boundary condition based on the use of residual free bubble functions an optimal order error bound is derived in the so called stability norm biggl eps nabla v 2 l 2 omega sum t h t conv cdot nabla v 2 l 2 t biggr 1 2 where ht denotes the diameter of element t in the subdivision of the computational domain

the dirichlet problem for singularly perturbed quasilinear elliptic equations assuming a c solution holland 9 used probabilistic methods to verify the regular and boundary layer expansions for 1 2 in case the fj arc independent of u and z e r van harten 141 proved the correctness of formal approximations for solutions of 1 2 with h 0 i n in case the boundary values are given on an n sphere howes 11 12 has obtained existence and asymptotic estimates for solutions of various boundary value problems associated with 1 2 in case the equation h z u 0 has a smooth solution a0 which satisfies certain stability conditions and such that the characteristic curves for the equation

singularly perturbed quadratically nonlinear dirichlet problems on etudie le probleme de dirichlet pour des equations elliptiques a perturbation singuliere de la forme e u a x u u u b x u u c x u dans e u on montre l existence de solutions pour e 0

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

uniform a priori estimates for singularly perturbed elliptic equations in multidimensions we consider a linear singularly perturbed elliptic equation on a bounded domain in mbox smallbbb r n this equation serves as a model for the interplay between the transport term and a relatively small viscosity term in some problems of mathematical physics we establish a priori estimates that hold uniformly in the small parameter from this one can prove an abstract a priori error estimate for the discretization with conforming finite elements for globally directed transport fields we establish a new type of anisotropic a priori estimate for the solution in the case of omega 0 1 2 we prove a priori estimates for certain higher order derivatives in isotropic as well as anisotropic norms in a subsequent paper we will apply these results to show uniform convergence of an exponentially fitted method on rectangular grids

a schwarz alternating procedure for singular perturbation problems we show that the schwarz alternating procedure offers a good algorithm for the numerical solution of singular perturbation problems provided the domain decomposition is properly designed to resolve the boundary and transition layers we give sharp estimates for the optimal position of the domain boundaries and study the convergence rates of the algorithm for various linear second order singular perturbation problems we report on implementation results for a turning point problem and a combustion problem

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

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

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

coercive singular perturbations ii reduction and convergence general coercive singular perturbations are reduced to regular ones using an algebra of singularly perturbed wiener hopf type operators high order asymptotic formulae also for non smooth data are indicated and sharp error estimates are established

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

projective free algebras of continuous functions on compact abelian groups abstract it is proved that the wiener algebra of functions on a connected compact abelian group whose bohr fourier spectra are contained in a fixed subsemigroup of the additive dual group is projective free the semigroup is assumed to contain zero and have the property that it does not contain both a nonzero element and its opposite the projective free property is proved also for the algebra of continuous functions with the same condition on their bohr fourier spectra as an application the connected components of the set of factorable matrices are described the proofs are based on a key result on homotopies of continuous maps on the maximal ideal spaces of the algebras under consideration

uniform error estimates for an exponentially fitted finite element method for singularly perturbed elliptic equations we consider a linear singularly perturbed elliptic equation on 0 1 2 in a recent paper w dorfler siam j numer anal to appear we proved uniform in the perturbation parameter estimates for solutions and an abstract a priori error estimate now we show for some examples that one can achieve uniform error estimates on a rectangular mesh using exponentially fitted finite elements these are constructed as tensor products from solutions of local one dimensional constant coefficient problems estimates are obtained in the l2 norm and assuming discrete stability also in the l infty norm

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

hp fem for reaction diffusion equations i robust exponential convergence a singularly perturbed reaction diffusion equation in two dimensions is considered we assume analyticity of the input data i e the boundary of the domain is an analytic curve and the right hand side is analytic we show that the hp version of the finite element method leads to robust exponential convergence provided that one layer of needle elements of width o p eps is inserted near the domain boundary that is the rate of convergence is o mbox rm exp b p and independent of the perturbation parameter eps additionally we show that the use of numerical quadrature for the evaluation of the stiffness matrix and the load vector retains the exponential rate of convergence in particular the spectral element method based on the use of a gauss lobatto quadrature rule with p 1 x p 1 points yields robust exponential convergence

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

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part ii internal stabilization in this second part of the paper a quadrat siam j control optim 40 2003 pp 266 299 we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions for internal stabilizability moreover we characterize all the integral domains a of siso stable plants such that every mimo plant defined by means of a transfer matrix whose entries belong to the quotient field k q a of a is internally stabilizable finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 s shankar and v r sule siam j control optim 30 1992 pp 11 30 v r sule siam j control optim 32 1994 pp 1675 1695 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

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

a rational spectral collocation method for third order singularly perturbed problems a new numerical method is developed for solving a class of third order singularly perturbed boundary value problems first of all the given problem is transformed into a system of two ordinary differential equations odes subject to suitable initial and boundary conditions then the rational spectral collocation method in barycentric form with sinh transformation is applied to solve the system of odes according to the asymptotic analysis the location and width of boundary layer of the given problem which are chosen as parameters in the sinh transformation can be determined ample numerical experiments are presented to illustrate the computational efficiency and accuracy of the our method

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

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 crosswind block iterative method for convection dominated problems this paper is on a block iterative algorithm for convection dominated equations the algorithm uses crosswind thin blocks in a block gauss seidel method the relaxation sweep is carried out successively along the downwind direction and exact solvers are used for the block systems this method is efficient for convection dominated problems discretized by monotone finite element finite difference schemes such as the edge average finite element method an optimal partitioning and ordering algorithm tarjan s algorithm is used to partition the nodes into crosswind blocks and order the blocks in the downwind direction the convergence of this block iterative method is analyzed and exponential convergence rates are proved for both one and two dimensional cases on both structured and unstructured meshes our empirical and analytical studies indicate that crosswind grouping is essential for the rapid convergence of the method and merely ordering the nodes along the downwind direction is not good enough some numerical examples are given to illustrate the effectiveness of the proposed algorithm for convection dominated problems and to compare it with other popular algorithms

preconditioning singularity perturbed elliptic and parabolic problems the goal of this paper is to investigate methods for preconditioning the system of linear equations that arises from a discretization of singularly perturbed multidimensional elliptic and parabolic problems for the problems under consideration it is important to investigate the conditioning of the resulting system as a function of certain parameters k and k 1 multiplying the wroth and first order derivative terms in the differential operator certain constraints relating k or k 1 to the gridsize h are imposed for reasons of stability and accuracy of the numerical solution the analysis involves estimating the condition number kappa l h 1 1 l h 2 in terms of k and k 1 where l h 1 and l h 2 are finite element discretizations of two singularly perturbed elliptic operators it is shown how to construct preconditioning operators such that the condition number of the preconditioned system of equations is bounded as the number of unknowns increases and at worst grows slowly as k

semilinear elliptic singular perturbation problems with nonuniform interior behavior pour un probleme elliptique de valeurs aux limites de la forme lu f u 0 dans bu h sur matano 1979 avait montre l existence d une solution intermediaire entre deux solutions stables quelconques ici on etudie ce type de solution pour le probleme de dirichlet e 2 u f u 0 dans u 0 sur ou e est un petit parametre est un domaine borne dans r n avec une frontiere du type c 2 pour un 0 1