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

locally constrained projections on grids a technique is formulated for projecting vector fields from one unstructured computational grid to another grid so that a constraint condition such as a conservation property holds at the cell or element level on the receiving grid the approach is based on ideas from constrained optimization and certain mixed or multiplier type finite element methods in which lagrange multipliers are introduced on the elements to enforce the constraint a theoretical analysis and estimates for the associated saddle point problem are developed and a new algorithm is proposed for efficient solution of the resulting discretized problem in the algorithm a reduced schur s complement problem is constructed for the multipliers and the projected velocity computation reduces to a post processing calculation in some instances the reduced system matrix can be factored so that repeated projections involve little more than forward and backward substitution sweeps numerical tests with an element of practical interest demonstrate optimal rate of convergence for the projected velocities and verify the local conservation property to expected machine precision a practical demonstration for environmental simulation of florida bay concludes the study copyright 2001 john wiley sons ltd

a cascadic multigrid algorithm for the stokes equations a variant of multigrid schemes for the stokes problem is discussed in particular we propose and analyse a cascadic version for the stokes problem the analysis of the transfer between the grids requires special care in order to establish that the complexity is the same as that for classical multigrid algorithms

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

computing point enclosures given a set of n rectangles in the plane the point enclosure query is the question to determine for any point p which rectangles of the set it is contained in it is the dual of the well known range query in computational geometry it is shown that the point enclosure query in the plane can be answered in 0 log n k time where k is the number of rectangles reported the solution makes use of a new data structure called the s tree the data structure can be generalized to obtain an efficient algorithm for the point enclosure problem in d dimensional space d 2

kronrod extensions with multiple nodes of quadrature formulas for fourier coefficients abstract we continue with analyzing quadrature formulas of high degree of precision for computing the fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials started recently by bojanov and petrova quadrature formulae for fourier coefficients j comput appl math 231 2009 378 391 and we extend their results construction of new gaussian quadrature formulas for the fourier coefficients of a function based on the values of the function and its derivatives is considered we prove the existence and uniqueness of kronrod extensions with multiple nodes of standard gaussian quadrature formulas with multiple nodes for several weight functions in order to construct some new generalizations of quadrature formulas for the fourier coefficients for the quadrature formulas for the fourier coefficients based on the zeros of the corresponding orthogonal polynomials we construct kronrod extensions with multiple nodes and highest algebraic degree of precision for this very desirable kind of extension there do not exist any results in the theory of standard quadrature formulas

calculation of gauss quadratures with multiple free and fixed knots algorithms are derived for the evaluation of gauss knots in the presence of fixed knots by modification of the jacobi matrix for the weight function of the integral simple gauss knots are obtained as eigenvalues of symmetric tridiagonal matrices and a rapidly converging simple iterative process based on the merging of free and fixed knots of quadratic convergence is presented for multiple gauss knots the procedures also allow for the evaluation of the weights of the quadrature corresponding to the simple gauss knots a new characterization of simple gauss knots as a solution of a partial inverse eigenvalue problem is derived

on the remainder term of gauss radau quadratures for analytic functions for analytic functions the remainder term of gauss radau quadrature formulae can be represented as a contour integral with a complex kernel we study the kernel on elliptic contours with foci at the points 1 and a sum of semi axes r 1 for the chebyshev weight function of the second kind starting from explicit expressions of the corresponding kernels the location of their maximum modulus on ellipses is determined the corresponding gautschi s conjecture from on the remainder term for analytic functions of gauss lobatto and gauss radau quadratures rocky mountain j math 21 1991 209 226 is proved

quadrature rules with multiple nodes for evaluating integrals with strong singularities we present a method based on the chakalov popoviciu quadrature formula of lobatto type a rather general case of quadrature with multiple nodes for approximating integrals defined by cauchy principal values or by hadamard finite parts as a starting point we use the results obtained by l gori and e santi cf on the evaluation of hilbert transforms by means of a particular class of turan quadrature rules numer algorithms 10 1995 27 39 quadrature rules based on s orthogonal polynomials for evaluating integrals with strong singularities oberwolfach proceedings applications and computation of orthogonal polynomials isnm 131 birkhauser basel 1999 pp 109 119 we generalize their results by using some of our numerical procedures for stable calculation of the quadrature formula with multiple nodes of gaussian type and proposed methods for estimating the remainder term in such type of quadrature formulae numerical examples illustrations and comparisons are also shown

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

gauss turan quadrature formulas asymptotics of weights first we derive asymptotics for the derivatives of s orthogonal polynomials and the derivatives of cauchy principal value integrals associated with s orthogonal polynomials with the help of these results asymptotics for the quadrature weights of gauss turan quadrature formulas are obtained so far asymptotics have been known with respect to the chebyshev weight function only

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

on a quadrature formula of gori and micchelli sparked by bojanov j comput appl math 70 1996 349 we provide an alternate approach to quadrature formulas based on the zeros of the chebyshev polynomial of the first kind for any weight function w introduced and studied in gori and micchelli math comp 65 1996 1567 thereby improving on their observations upon expansion of the divided differences we obtain explicit expressions for the corresponding cotes coefficients in gauss turan quadrature formulas for i f w 1 1f x w x dx and i ft n w for a gori micchelli weight function it is also interesting to mention what has been neglected for about 30 years by the literature is that as a consequence of expansion of the divided differences in the special case when w x 1 1 x 2 the solution of the famous turan s problem 26 raised in 1980 was in fact implied by a result of micchelli and rivlin ibm j res develop 16 1972 372 in 1972 some concluding comments are made in the final section

dynamization of decomposable searching problems yielding good worsts case bounds a technique is described for dynamizing data structures of common types of decomposable searching problems yielding good worst case bounds rather than merely good average bounds on all operations to be performed on the resulting structures

on weight functions which admit explicit gauss tura n quadrature formulas the main purpose of this paper is the construction of explicit gauss turan quadrature formulas they are relative to some classes of weight functions which have the peculiarity that the corresponding s orthogonal polynomials of the same degree are independent of s these weights too are introduced and discussed here moreover highest precision quadratures for evaluating fourier chebyshev coefficients are given

efficient splitting and merging algorithms for order decomposable problems let s be a set whose items are sorted with respect to d 1 total orders 1 d and which is subject to dynamic operations such as insertions of a single item deletions of a single item split and concatenate operations performed according to any chosen order i 1 i d this generalizes to dimension d 1 the notion of concatenable data structures such as the 2 3 trees which support splits and concatenates under a single total order the main contribution of this paper is a general and novel technique for solving order decomposable problems on s which yields new and efficient concatenable data structures for dimension d 1 by using our technique we maintain s with the time bounds o logn for the insertion or the deletion of a single item where n is the number of items currently in s n1 1 d for splits and concatenates along any order and for rectangular range queries the space required is o n we provide several applications of our technique namely we present new multidimensional data structures implementing two dimensional priority queues two dimensional search trees and concatenable interval trees these data structures allow us to improve many previously known results on decomposable problems under split and concatenate operations such as membership query minimum weight item range query convex hulls and voronoi diagrams

a moving mesh finite element method for the two dimensional stefan problems an r adaptive moving mesh method is developed for the numerical solution of an enthalpy formulation of two dimensional heat conduction problems with a phase change the grid is obtained from a global mapping of the physical to the computational domain which is designed to cluster mesh points around the interface between the two phases of the material the enthalpy equation is discretised using a semiimplicit galerkin finite element method using linear basis functions the moving finite element method is applied to problems where the phase front is cusp shaped and where the interface changes topology

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

quadrature rules using first derivatives for oscillatory integrands we consider the integral of a function y x i y x 11 y x dx and its approximation by a quadrature rule of the form qn y x k 1n wky xk k 1n ky xk i e by a rule which uses the values of both y and its derivative at nodes of the quadrature rule we examine the cases when the integrand is either a smooth function or an dependent function of the form y x f1 x sin x f2 x cos x with smoothly varying f1 and f2 in the latter case the weights wk and k are dependent we establish some general properties of the weights and present some numerical illustrations

projection and iteration in adaptive finite element refinement a local elliptic projection is introduced to provide improved local approximations to the finite element solution of boundary value problems the approach yields a composite global approximation that is more accurate and provides a good starting value for iterative solution methods the procedure is particularly effective when used in conjunction with adaptive mesh refinement as illustrated in numerical studies involving two point problems of boundary layer and interior layer type

error bounds for gauss turan quadrature formulae of analytic functions we study the kernels of the remainder term rn s f of gauss thran quadrature formulas 11f t w t dt ai vf i v rn s f n n s n0 for classes of analytic functions on elliptical contours with foci at 1 when the weight w is one of the special jacobi weights w t 1 t 1 t 1 2 1 2 s 1 2 1 2 s 1 2 s 1 2 we investigate the location on the contour where the modulus of the kernel attains its maximum value some numerical examples are included

a note on the bounds of the error of gauss turan type quadratures this note is concerned with estimates for the remainder term of the gauss turan quadrature formula r n s f 1 1w t f t dt n 1n i 02sa i nf i t n where w t u n 1 t n 21 t 2 is the gori michelli weight function with u n 1 t denoting the n 1 th degree chebyshev polynomial of the second kind and f is a function analytic in the interior of and continuous on the boundary of an ellipse with foci at the points 1 and sum of semiaxes r 1 the present paper generalizes the results in g v milovanovic m m spalevic bounds of the error of gauss turan type quadratures j comput appl math 178 2005 333 346 which is concerned with the same problem when s 1

the 4 triangles longest side partition of triangles and linear refinement algorithms in this paper we study geometrical properties of the iterative 4 triangles longest side partition of triangles and of a 3 triangles partition as well as practical algorithms based on these partitions used both directly for the triangulation refinement problem and as a basis for point insertion strategies in delaunay refinement algorithms the 4 triangles partition is obtained by joining the midpoint of the longest side with the opposite vertex and the midpoints of the two remaining sides by means of simple geometrical arguments we show that the iterative partition of obtuse triangles systematically improves the triangles while they remain obtuse in the following sense the sequence of smallest angles monotonically increases while the sequence of largest angles monotonically decreases in an amount at least equal to the smallest angle of each iteration this allows us to improve the known bound on the smallest angle without making use of previous results and to obtain a better a priori bound on the number of similarly distinct triangles as a function of the geometry of the initial triangle numerical evidence showing that the practical behavior of the 4 triangles partition is in complete agreement with this theory is included a 4 triangles refinement algorithm is also discussed and illustrated furthermore we show that the time cost of the algorithm is linear independently of the size of the triangulation

bounds of the error of gauss turan type quadratures we consider the remainder term of the gauss turan quadrature formulae rn s f 11w t f t dt v 1n i 02sai vf i v for analytic functions in some region of the complex plane containing the interval 1 1 in its interior the remainder term is presented in the form of a contour integral over confocal ellipses or circles a strong error analysis is given for the case with a generalized class of weight functions introduced recently by gori and micchelli also we discuss a general case with an even weight function defined on 1 1 numerical results are included

some aspects of adaptive grid technology related to boundary and interior layers we consider the use of adaptive mesh strategies for solution of problems exhibiting boundary and interior layer solutions as the presence of these layer structures suggests reliable and accurate solution of this class of problems using finite difference finite volume or finite element schemes requires grading the mesh into the layers and due attention to the associated algorithms when the nature and structure of the layer is known mesh grading can be achieved during the grid generation by specifying an appropriate grading function however in many applications the location and nature of the layer behavior is not known in advance consequently adaptive mesh techniques that employ feedback from intermediate grid solutions are an appealing approach in this paper we provide a brief overview of the main adaptive grid strategies in the context of problems with layers associated error indicators that guide the refinement feedback control grid optimization process are also covered and there is a brief commentary on the supporting data structure requirements some current issues concerning the use of stabilization in conjunction with adaptive mesh refinement amr the question of pollution effects in computation of local error indicators the influence of nonlinearities and the design of meshes for targeted optimization of specific quantities are considered the application of amr for layer problems is illustrated by means of case studies from semiconductor device transport drift diffusion nonlinear reaction diffusion layers due to surface capillary effects and shockwaves in compressible gas dynamics

consistency stability a priori and a posteriori errors for petrov galerkin methods applied to nonlinear problems in an abstract framework we present a formalism which specifies the notions of consistency and stability of petrov galerkin methods used to approximate nonlinear problems which are in many practical situations strongly nonlinear elliptic problems this formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems this theory is illustrated with the example div k u u c u f in a two dimensional domain with dirichlet boundary conditions

new quadrature formulas based on the zeros of the chebyshev polynomials of the second kind abstract the aim of this work is to construct a new quadrature formula based on the divided differences of the integrand at points 1 1 and the zeros of the nth chebyshev polynomial of the second kind the interesting thing is that this quadrature rule is closely related to the well known gauss turan quadrature formula and includes a recent result obtained by a k varma and e landau as a special case

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

quadratures with multiple nodes power orthogonality and moment preserving spline approximation quadrature formulas with multiple nodes power orthogonality and some applications of such quadratures to moment preserving approximation by defective splines are considered an account on power orthogonality s and orthogonal polynomials and generalized gaussian quadratures with multiple nodes including stable algorithms for numerical construction of the corresponding polynomials and cotes numbers are given in particular the important case of chebyshev weight is analyzed finally some applications in moment preserving approximation of functions by defective splines are discussed