a posteriori error estimation based on potential and flux reconstruction for the heat equation we derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous galerkin various finite volume and mixed finite element methods in space and the backward euler scheme in time extensions to conforming and nonconforming finite element spatial discretizations are also outlined our estimates are based on a h 1 conforming reconstruction of the potential continuous and piecewise affine in time and a locally conservative mathbf h mathrm div conforming reconstruction of the flux piecewise constant in time they yield a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative local in time lower bounds are also derived for nonconforming methods on time varying meshes the lower bounds require a mild parabolic type constraint on the meshsize

guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems abstract we derive a posteriori error estimates for a class of second order monotone quasi linear diffusion type problems approximated by piecewise affine continuous finite elements our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual as well as a global error lower bound up to a generic constant independent of the nonlinear operator they are thus fully robust with respect to the nonlinearity thanks to the choice of the error measure they are also locally efficient albeit in a different norm and hence suitable for adaptive mesh refinement moreover they allow to distinguish estimate separately and compare the discretization and linearization errors hence the iterative newton raphson fixed point linearization can be stopped whenever the linearization error drops to the level at which it does not affect significantly the overall error this can lead to important computational savings as performing an excessive number of unnecessary linearization iterations can be avoided a strategy combining the linearization stopping criterion and adaptive mesh refinement is proposed and numerically tested for the p laplacian

explicit and averaging a posteriori error estimates for adaptive finite volume methods local mesh refining algorithms known from adaptive finite element methods are adopted for locally conservative and monotone finite volume discretizations of boundary value problems for steady state convection diffusion reaction equations the paper establishes residual type explicit error estimators and averaging techniques for a posteriori finite volume error control with and without upwind in global h1 and l2 norms reliability and efficiency are verified theoretically and confirmed empirically with experimental support for the superiority of the suggested adaptive mesh refining algorithms over uniform mesh refining a discussion of adaptive computations in the simulation of contaminant concentration in a nonhomogeneous water reservoir concludes the paper

piecewise polynomial collocation for boundary integral equations this paper considers the numerical solution of boundary integral equations of the second kind for laplace s equation delta u 0 on connected regions d in bf r 3 with boundary s the boundary s is allowed to be smooth or piecewise smooth and we let delta k 1 leq k leq n be a triangulation of s the numerical method is collocation with approximations which are piecewise quadratic in the parametrization variables leading to a numerical solution u n superconvergence results for u n are given for s a smooth surface and for a special type of refinement strategy for the triangulation we show that u u n is o delta 4 log delta at the collocation node points with delta being the mesh size for delta k error analyses are given are given for other quantities and an important error analysis is given for the approximation of s by piecewise quadratic interpolation on each triangular element with s either smooth or piecewise smooth the convergence result we

a posteriori error estimates for combined finite volume finite element discretizations of reactive transport equations on nonmatching grids abstract we derive in this paper guaranteed and fully computable a posteriori error estimates for vertex centered finite volume type discretizations of transient linear convection diffusion reaction equations our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew symmetric part of the differential operator lower bounds global in space but local in time are also derived these lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting on the basis of the derived estimates we propose an adaptive algorithm which enables to automatically achieve a user given relative precision this algorithm also leads to efficient calculations as it balances the time and space error contributions as an example we apply our estimates to the combined finite volume finite element scheme including such features as use of mass lumping for the time evolution or reaction terms of upwind weighting for the convection term and discretization on nonmatching meshes possibly containing nonconvex and non star shaped elements a collection of numerical experiments illustrates the efficiency of our estimates and the use of the space time adaptive algorithm

fractional differential equations and lyapunov functionals abstract we consider a scalar fractional differential equation write it as an integral equation and construct several lyapunov functionals yielding qualitative results about the solution it turns out that the kernel is convex with a singularity and it is also completely monotone as is the resolvent kernel while the kernel is not integrable the resolvent kernel is positive and integrable with an integral value of one these kernels give rise to essentially different types of lyapunov functionals it is to be stressed that the lyapunov functionals are explicitly given in terms of known functions and they are differentiated using leibniz s rule the results are readily accessible to anyone with a background of elementary calculus

a posteriori error estimations of some cell centered finite volume methods for diffusion convection reaction problems this paper presents an a posteriori residual error estimator for diffusion convection reaction problems approximated by some cell centered finite volume methods on isotropic or aniso tropic meshes in rd d 2 or 3 for that purpose we built a reconstructed approximation which is an appropriate interpolant of the finite volume solution the error is then the difference between the exact solution and this interpolant the residual error estimator is based on the jump of the normal derivative of the interpolant we then prove the equivalence between the energy norm of the error and the residual error estimator some numerical tests confirm our theoretical results

trigonometric hermite wavelet approximation for the integral equations of second kind with weakly singular kernel this paper is concerned with a trigonometric hermite wavelet galerkin method for the fredholm integral equations with weakly singular kernel the kernel function of this integral equation considered here includes two parts a weakly singular kernel part and a smooth kernel part the approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric hermite wavelet constructed by e quak trigonometric wavelets for hermite interpolation math comp 65 1996 683 722 are developed the use of trigonometric hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix it is shown that we only need to compute and store o n entries for the weakly singular kernel representation matrix with dimensions n 2 which can reduce the whole computational cost and storage expense the computational schemes of the resulting matrix elements are provided for the weakly singular kernel function furthermore the convergence analysis is developed for the trigonometric wavelet method in this paper

wavelet galerkin schemes for boundary integral equations implementation and quadrature in the present paper we consider the fully discrete wavelet galerkin scheme for the fast solution of boundary integral equations in three dimensions it produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that stays proportional to the number of unknowns we focus on algorithmical details of the scheme in particular on numerical integration of relevant matrix coefficients we illustrate the proposed algorithm by numerical results

a meshless method for solving nonlinear two dimensional integral equations of the second kind on non rectangular domains using radial basis functions with error analysis in this paper we present a numerical method for solving two dimensional nonlinear fredholm integral equations of the second kind on a non rectangular domain the method utilizes radial basis functions rbfs constructed on scattered points as a basis in the discrete collocation method the proposed scheme is meshless since it does not need any domain element and so it is independent of the geometry of the domain the method reduces the solution of the two dimensional nonlinear integral equation to the solution of a nonlinear system of algebraic equations error analysis is presented for this method finally numerical examples are included to show the validity and efficiency of the new technique

a note on constant free a posteriori error estimates in this note we look at constant free a posteriori error estimates from a different perspective we show that they can be interpreted as an alternative way of expressing the residual of a finite element approximation and thus fit into the same framework as other a posteriori error estimates such as residual error indicators our approach also reveals that when applied to singularly perturbed reaction diffusion or convection diffusion problems constant free a posteriori error estimates will not be fully robust unless extra measures are taken

guaranteed and robust discontinuous galerkin a posteriori error estimates for convection diffusion reaction problems we propose and study a posteriori error estimates for convection diffusion reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior penalty discontinuous galerkin methods our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction we first derive locally computable estimates for the error measured in the energy semi norm these estimates are evaluated using h div w conforming diffusive and convective flux reconstructions thereby extending the previous work on pure diffusion problems the resulting estimates are semi robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local peclet and damkohler numbers fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy semi norm a dual norm of the skew symmetric part of the differential operator and a suitable contribution of the interelement jumps of the discrete solution numerical experiments are presented to illustrate the theoretical results

projection and iterated projection methods for nonliear integral equations consider the nonlinear operator equation x mathcal k x with mathcal k a completely continuous mapping of a domain in the banach space mathcal x into mathcal x and let x denote an isolated fixed point of mathcal k let mathcal k n n geqq 1 denote a sequence of finite dimensional approximating subspaces and let p n be a projection of mathcal x onto mathcal x n the projection method for solving x mathcal k x given by x n p n mathcal k x n and the iterated projection solution is defined as tilde x n mathcal k x n we analyze the convergence of x n and tilde x n to x giving a general analysis that includes both the galerkin and collocation methods a more detailed analysis is then given for a large class of urysohn integral operators in one variable showing the superconvergence of tilde x n to x

inverse source problem in an oblate spheroidal geometry the canonical inverse source problem of reconstructing an unknown source whose region of support is describable as a spheroidal oblate or prolate volume from knowledge of the far field radiation pattern it generates is formulated and solved within the framework of the inhomogeneous scalar helmholtz equation via a linear inversion framework in hilbert spaces particular attention is paid to the analysis and computer illustration of flat aperture like sources whose support is approximated by an oblate spheroidal volume

analysis of a finite volume element method for the stokes problem in this paper we propose a stabilized conforming finite volume element method for the stokes equations on stating the convergence of the method optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations a superconvergence result is also derived by using a postprocessing projection method in particular the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries finally some numerical experiments that confirm the predicted behavior of the method are provided

a spectral galerkin method for a boundary integral equation on considere une equation integrale liee au probleme bidimensionnel de dirichlet resolu par la methode du potentiel de simple couche on analyse une methode de galerkin spectrale adaptee au cas des donnees aux limites et d un domaine reguliers on accelere la convergence dans les espaces de sobolev en utilisant des polynomes trigonometriques

numerical solution to the unsteady two dimensional schrodinger equation using meshless local boundary integral equation method a meshless local boundary integral equation lbie method is proposed to solve the unsteady two dimensional schrodinger equation the method is based on the lbie with moving least squares mls approximation for the mls approximation nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables a time stepping method is employed to deal with the time derivative an efficient method for dealing with singular domain integrations that appear in the discretized equations is presented finally numerical results are considered for some examples to demonstrate the accuracy efficiency and high rate of convergence of this method copyright 2008 john wiley sons ltd

robust a posteriori error estimates for nonstationary convection diffusion equations we consider discretizations of convection dominated nonstationary convection diffusion equations by a stable theta schemes in time and conforming finite elements in space on locally refined isotropic meshes for these discretizations we derive a residual a posteriori error estimator the estimator yields upper bounds on the error which are global in space and time and lower bounds that are global in space and local in time the error estimates are fully robust in the sense that the ratio between upper and lower bounds is uniformly bounded in time does not depend on any step size in space or time nor on any relation between these both and is uniformly bounded with respect to the size of the convection moreover the estimates are uniform with respect to the size of the zero order reaction term and also hold for the limit case of vanishing reaction

solving a class of two dimensional linear and nonlinear volterra integral equations by the differential transform method in this paper we will give some results for developing the two dimensional differential transform tddt for double integrals then the tddt method will be developed for solving a class of two dimensional linear and nonlinear volterra integral equations we also give some examples to demonstrate the accuracy of the method

adaptive regularization linearization and discretization and a posteriori error control for the two phase stefan problem we consider in this paper the time dependent two phase stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward euler temporal discretizations regularization of the enthalpy temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual as well as on the l2 l2 error of the temperature and the l2 h1 error of the enthalpy moreover they allow to distinguish the space time regularization and linearization error components an adaptive algorithm is proposed which ensures computational savings through the online choice of a sufficient regularization parameter a stopping criterion for the linearization iterations local space mesh refinement time step adjustment and equilibration of the spatial and temporal errors we also prove the efficiency of our estimate our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization as an example we apply it to the vertex centered finite volume finite element with mass lumping and quadrature and newton methods numerical results illustrate the effiectiveness of our estimates and the performance of the adaptive algorithm

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

gauss type quadratures for weakly singular integrals and their application to fredholm integral equations of the second kind in this paper we establish gauss type quadrature formulas for weakly singular integrals an application of the quadrature scheme is given to obtain numerical solutions of the weakly singular fredholm integral equation of the second kind we call this method a discrete product integration method since the weights involved in the standard product integration method are computed numerically

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

adaptive streamline diffusion finite element methods for stationary convection diffusion problems adaptive finite element methods for stationary convectiondiffusion problems are designed and analyzed the underlying discretization scheme is the shock capturing streamline diffusion method the adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed a priori error estimates are used to show that the algorithms are efficient in a certain sense

on the use of boundary conditions for variational formulations arising in financial mathematics the general intractability of derivative security valuation models to present techniques both analytic and numerical arguably remains one of the preeminant problem of mathematical finance it is the focus of this paper to examine the applicability of a promising recent development namely radial basis functions rbf to the problem of option valuation a black scholes framework is considered for american and european options written on a one and two risky assets the performance of rbf and finite differencing algorithms are examined with respect to artificial boundary conditions computational domain domain decomposition and mesh scaling

implementation of meshless lbie method to the 2d non linear sg problem in this paper the meshless local boundary integral equation lbie method for numerically solving the non linear two dimensional sine gordon sg equation is developed the method is based on the lbie with moving least squares mls approximation for the mls nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables the approximation functions are constructed entirely using a set of scattered nodes and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes a time stepping method is employed to deal with the time derivative and a simple predictor corrector scheme is performed to eliminate the non linearity a brief discussion is outlined for numerical integrations in the proposed algorithm some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped sg equation is investigated the final numerical results confirm the ability of method to deal with the unsteady non linear problems in large domains copyright 2009 john wiley sons ltd

error estimates for a combined finite volume finite element method for nonlinear convection diffusion problems the subject of this paper is the analysis of error estimates of the combined finite volume finite element fv fe method for the numerical solution of a scalar nonlinear conservation law equation with a diffusion term nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume mesh dual to a triangular grid whereas the diffusion term is discretized by piecewise linear conforming triangular finite elements under the assumption that the exact solution possesses some regularity properties and the triangulations are of a weakly acute type with the aid of the discrete maximum principle and a priori estimates error estimates of the method are proved

a contraction mapping proof of the smooth dependence on parameters of solutions to volterra integral equations abstract we consider the linear volterra equation x t a t 0 t k t s x s d s and suppose that the kernel k and forcing function a depend on some parameters r d we prove that under suitable conditions the solutions depend on as smoothly the functions a and k the proof is based on the contraction mapping principle and the variational equation though our conditions are not the most generally possible they nonetheless include many important examples

numerical solution of the nonlinear fredholm integral equations by positive definite functions a numerical method for solving the nonlinear fredholom integral equations is presented the method is based on interpolation by radial basis functions rbf to approximate the solution of the fredholm nonlinear integral equations several examples are given and numerical examples are presented to demonstrate the validity and applicability of the method

a liapunov functional for a singular integral equation abstract we consider a scalar integral equation x t a t 0 t c t s g s x s d s where a l 2 0 while c t s has a significant singularity but is convex when t s 0 we construct a liapunov functional and show that g t x t a t l 2 0 and that x t a t 0 pointwise as t small perturbations are also added to the kernel in addition we consider both infinite and finite delay problems this paper offers a first step toward treating discontinuous kernels with liapunov functionals

a meshless discrete collocation method for the numerical solution of singular logarithmic boundary integral equations utilizing radial basis functions abstract the main intention of the current paper is to describe a scheme for the numerical solution of boundary integral equations of the second kind with logarithmic singular kernels these types of integral equations result from boundary value problems of laplace s equations with linear robin boundary conditions the method approximates the solution using the radial basis function rbf expansion with polynomial precision in the discrete collocation method the collocation method for solving logarithmic boundary integral equations encounters more difficulties for computing the singular integrals which cannot be approximated by the classical quadrature formulae to overcome this problem we utilize the non uniform composite gauss legendre integration rule and employ it to estimate the singular logarithm integrals appeared in the method since the scheme is based on the use of scattered points spread on the analyzed domain and does not need any domain elements we can call it as the meshless discrete collocation method the new algorithm is successful and easy to solve various types of boundary integral equations with singular kernels we also provide the error estimate of the proposed method the efficiency and accuracy of the new approach are illustrated by some numerical examples

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

the numerical solution of the non linear integro differential equations based on the meshless method this article investigates the numerical solution of the nonlinear integro differential equations the numerical scheme developed in the current paper is based on the moving least square method the moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data it consists of a local weighted least square fitting valid on a small neighborhood of a point and only based on the information provided by its n closet points hence the method is a meshless method and does not need any background mesh or cell structures the error analysis of the proposed method is provided the validity and efficiency of the new method are demonstrated through several tests

finite volume element approximation of an inhomogeneous brusselator model with cross diffusion this paper is concerned with the study of pattern formation for an inhomogeneous brusselator model with cross diffusion modeling an autocatalytic chemical reaction taking place in a three dimensional domain for the spatial discretization of the problem we develop a novel finite volume element fve method associated to a piecewise linear finite element approximation of the cross diffusion system we study the main properties of the unique equilibrium of the related dynamical system a rigorous linear stability analysis around the spatially homogeneous steady state is provided and we address in detail the formation of turing patterns driven by the cross diffusion effect in addition we focus on the spatial accuracy of the fve method and a series of numerical simulations confirm the expected behavior of the solutions in particular we show that depending on the spatial dimension the magnitude of the cross diffusion influences the selection of spatial patterns

a fast numerical solution for a second kind boundary integral equation with a logarithmic kernel an iterative quadrature method is presented and analyzed for a second kind fredholm integral equation with a logarithmic kernel which arises from the boundary integral equation reformulation of some boundary value problems for the two dimensional helmholtz equation the method combines a quadrature method for discretizing the integral equation with a preconditioned iterative method for solving the resulting dense and nonsymmetric linear system the quadrature method has a polynomial or exponential rate of convergence which can be retained by the preconditioned iterative method with only o n 2 arithmetic operations

bernstein series solutions of multidimensional linear and nonlinear volterra integral equations with fractional order weakly singular kernels abstract this paper proposes a quadrature method based on multi variate bernstein polynomials the method is used to solve multidimensional volterra integral equations with weakly singular kernels firstly we use multi variate bernstein polynomials to approximate the unknown function of an equation then a discrete function equation can be obtained by substituting the approximate solution into the equation secondly the discrete function system is transformed into an algebra equation system by using some discrete points we can perform the integral operations without discrete kernel function and the weakly singular integrals can be calculated directly by using quadrature method so the method is easy to implement thirdly we prove the existence and uniqueness of the solution of the approximate equation as well as the error analysis of the proposed method six numerical examples are given to illustrate the efficiency of this method

solution of a class of two dimensional integral equations the two dimensional integral equation 1 d r r ds f r0 0 defined on a circular disk d r0 a 0 0 2 is considered in the present paper here r in the kernel denotes the distance between two points p r and p0 r0 0 in d and 0 2 or 2 4 based on some known results of bessel functions integral representations of the kernel are established for 0 2 and 2 4 respectively and employed to solve the corresponding two dimensional integral equation the solutions of the weakly singular integral equation for 0 2 and of the hypersingular integral equation for 2 4 are obtained respectively

residual flux based a posteriori error estimates for finite volume and related locally conservative methods we derive in this paper a posteriori error estimates for discretizations of convection diffusion reaction equations in two or three space dimensions our estimates are valid for any cell centered finite volume scheme and in a larger sense for any locally conservative method such as the mimetic finite difference covolume and other we consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces we allow for the cases of inhomogeneous and anisotropic diffusion dispersion tensors and of convection dominance the estimates are established in the energy semi norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type they are fully computable all occurring constants are evaluated explicitly and locally efficient give a local lower bound on the error times an efficiency constant so that they can serve both as indicators for adaptive refinement and for the actual control of the error they are semi robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local peclet number gets sufficiently small numerical experiments confirm their accuracy

convergence analysis for a conformal discretization of a model for precipitation and dissolution in porous media in this paper we discuss the numerical analysis of an upscaled core scale model describing the transport precipitation and dissolution of solutes in a porous medium the particularity lies in the modeling of the reaction term especially the dissolution term which has a multivalued character we consider the weak formulation for the upscaled equation and provide rigorous stability and convergence results for both the semi discrete time discretization and the fully discrete schemes in doing so compactness arguments are employed

richardson extrapolation of iterated discrete galerkin solution for two dimensional fredholm integral equations in this paper we study the numerical solution of two dimensional fredholm integral equation by discrete galerkin and iterated discrete galerkin method we are able to derive an asymptotic error expansion of the iterated discrete galerkin solution this expansion covers arbitrarily high powers of the discretization parameters if the solution of the integral equation is smooth the expansion gives rise to richardson type extrapolation schemes which rapidly improve the original rate of the convergence numerical experiments confirm our theoretical results

a meshless discrete galerkin mdg method for the numerical solution of integral equations with logarithmic kernels abstract this paper describes a computational method for solving fredholm integral equations of the second kind with logarithmic kernels the method is based on the discrete galerkin method with the shape functions of the moving least squares mls approximation constructed on scattered points as basis the mls methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least square polynomial fitting the numerical scheme developed in the current paper utilizes the non uniform gauss legendre quadrature rule for approximating logarithm like singular integrals and so reduces the solution of the logarithmic integral equation to the solution of a linear system of algebraic equations the proposed method is meshless since it does not require any background mesh or domain elements the error analysis of the method is provided the scheme is also applied to a boundary integral equation which is a reformulation of a boundary value problem of laplace s equation with linear robin boundary conditions finally numerical examples are included to show the validity and efficiency of the new technique

numerical solution of nonlinear volterra fredholm hammerstein integral equations via collocation method based on radial basis functions abstract a numerical technique based on the spectral method is presented for the solution of nonlinear volterra fredholm hammerstein integral equations this method is a combination of collocation method and radial basis functions rbfs with the differentiation process drbf using zeros of the shifted legendre polynomial as the collocation points different applications of rbfs are used for this purpose the integral involved in the formulation of the problems are approximated based on legendre gauss lobatto integration rule the results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme

adaptive finite elements for a linear parabolic problem a posteriori error estimates for the heat equation in two space dimensions are presented a classical discretization is used euler backward in time and continuous piecewise linear triangular finite elements in space the error is bounded above and below by an explicit error estimator based on the residual numerical results are presented for uniform triangulations and constant time steps the quality of our error estimator is discussed an adaptive algorithm is then proposed successive delaunay triangulations are generated so that the estimated relative error is close to a preset tolerance again numerical results demonstrate the efficiency of our approach c 1998 elsevier science s a all rights reserved

numerical solution of the second kind integral equations using radial basis function networks in this paper a new method using radial basis function rbf networks is presented for solving the linear second kind integral equations of fredholm and volterra types this method employs a growing neural network as the approximate solution of the integral equations whose the activation functions are rbfs the parameters weights centers and widths of the approximate solution are adjusted by using an unconstrained optimization problem numerical results show that our method has the potentiality to become an efficient approach for solving integral equations

a discrete collocation type method for hammerstein equations in recent papers we have considered the numerical solution of the hammerstein equation y t f t int a b k t s g s y s ds quad t in a b by a method that first applies the standard collocation procedure to an equivalent equation for z t g t y t and then obtains an approximation to y by use of the equation y t f t int a b k t s z s ds quad t in a b we study here a discretized version of the above method this arises when numerical quadrature is used to approximate the definite integrals occurring in the said method we consider the use of interpolatory quadrature rules and we seek the discrete collocation approximation to z in certain piecewise polynomial function spaces our principal result gives the precision required of a quadrature rule to guarantee the best possible super convergence rate for the discrete approximation to y in the case where the kernel k is sufficiently smooth this rate is of the same order as the rate for an approximation to y obtain

universal quadratures for boundary integral equations on two dimensional domains with corners we describe the construction of a collection of quadrature formulae suitable for the efficient discretization of certain boundary integral equations on a very general class of two dimensional domains with corner points the resulting quadrature rules allow for the rapid high accuracy solution of dirichlet boundary value problems for laplace s equation and the helmholtz equation on such domains under a mild assumption on the boundary data our approach can be adapted to other boundary value problems and certain aspects of our scheme generalize to the case of surfaces with singularities in three dimensions the performance of the quadrature rules is illustrated with several numerical examples

a galerkin boundary node method and its convergence analysis the boundary node method bnm exploits the dimensionality of the boundary integral equation bie and the meshless attribute of the moving least square mls approximations however since mls shape functions lack the property of a delta function it is difficult to exactly satisfy boundary conditions in bnm besides the system matrices of bnm are non symmetric a galerkin boundary node method gbnm is proposed in this paper for solving boundary value problems in this approach an equivalent variational form of a bie is used for representing the governing equation and the trial and test functions of the variational formulation are generated by the mls approximation as a result boundary conditions can be implemented accurately and the system matrices are symmetric total details of numerical implementation and error analysis are given for a general bie taking the dirichlet problem of laplace equation as an example we set up a framework for error estimates of gbnm some numerical examples are also given to demonstrate the efficacity of the method

an explicit kernel split panel based nystrom scheme for integral equations on axially symmetric surfaces a high order accurate explicit kernel split panel based fourier nystrom dis cretization scheme is developed for integral equations associated with the helm holtz equation in axially symmetric domains extensive incorporation of ana lytic information about singular integral kernels and on the y computation of nearly singular quadrature rules allow for very high achievable accuracy also in the evaluation of elds close to the boundary of the computational domain

on the solution of an integral equation arising in potential problems for circular and elliptic disks this paper deals with a two dimensional fredholm integral equation of the first kind over the circular disk s with a kernel of the form g theta bf r bf r bf r bf r in s where b is the angle between bf r bf r and some reference direction by expansions in fourier series and in series involving legendre functions and by use of a new closed form result for a legendre function integral the integral equation is reduced to a system of linear equations for the expansion coefficients it is shown that the system has a unique solution because of the toeplitz structure of the system matrix as an application the electrostatic potential problem for a charged elliptic disk is discussed

galerkin trigonometric wavelet methods for the natural boundary integral equations a simply supported boundary value problem of biharmonic equations in the unit disk is reduced into an equivalent second kind natural boundary integral equation nbie with hypersingular kernel in the sense of hadamard finite part trigonometric hermite interpolatory wavelets introduced by quak math comput 65 1996 683 722 as trial functions are used to its galerkin discretization with 2 j 1 degrees of freedom on the boundary it is proved that the stiffness matrix is a block diagonal matrix and its diagonal elements are some symmetric and block circulant submatrices the simple computational formulae of the entries in stiffness matrix are obtained these show that we only need to compute 2 2 j j 1 elements of a 2 j 2x2 j 2 stiffness matrix the error estimates for the approximation solutions are established finally numerical examples are given

wavelet approximations for first kind boundary integral equations on polygons an elliptic boundary value problem in the interior or exterior of a polygon is transformed into an equivalent first kind boundary integral equation its galerkin discretization with n degrees of freedom on the boundary with spline wavelets as basis functions is analyzed a truncation strategy is presented which allows to reduce the number of nonzero elements in the stiffness matrix from o n 2 to o n log n entries the condition numbers are bounded independently of the meshwidth it is proved that the compressed scheme thus obtained yields in o n log n 2 operations approximate solutions with the same asymptotic convergence rates as the full galerkin scheme in the boundary energy norm as well as in interior points numerical examples show the asymptotic error analysis to be valid already for moderate values of n

a finite volume formulation for transient convection and diffusion equations with unstructured distorted grids and its applications in fluid flow simulations with a collocated variable arrangement abstract this paper describes a finite volume method for simulating transport processes governed by convection diffusion type equations the formulation is based on a cell centred unstructured grid with an edge based data structure discretisation is independent of control volume or cell shape by using a surface vector decomposition at the midpoint of the interface between cells along with a deferred correction approach any cross diffusion due to grid skewness can be readily accounted for when discretising the diffusive flux for modelling fluid flow processes a collocated arrangement of variables is employed so that a single coefficient matrix applies for the momentum equations of each velocity component to avoid checkerboard oscillation arising from pressure velocity decoupling occurring under the collocated variable arrangement when a pressure based solution algorithm is employed a novel pseudo flux interpolation method is proposed for unstructured grids ensuring that the solution is both under relaxation factor and time step for transient calculation independent the methodology can be formulated within a framework whereby either a coupled or a decoupled solution algorithm can be employed the features and advantages of the method are demonstrated by solution of the navier stokes equations for two benchmark flow problems

a radial basis function hermite finite difference approach to tackle cash or nothing and asset or nothing options abstract a novel local meshfree method is proposed for simulating cash or nothing and asset or nothing options at which the initial condition is discontinuous the novelty of the scheme is in the use of hermite radial basis function rbf interpolation on set of local nodes we also construct the rbf generated hermite finite difference hfd formulas and subsequently obtain their associated differentiation matrices dms the developed accuracy equipped with a shift algorithm for smoothing the discontinuous initial conditions are illustrated with a series of financial tests

robust a posteriori error estimates for stationary convection diffusion equations we analyze a posteriori error estimators for finite element discretizations of convection dominated stationary convection diffusion equations using locally refined isotropic meshes the estimators are based on either the evaluation of local residuals or the solution of discrete local problems with dirichlet or neumann boundary conditions all estimators yield global upper and lower bounds for the error measured in a norm that incorporates the standard energy norm and a dual norm of the convective derivative they are fully robust in the sense that the ratio of the upper and lower bounds is uniformly bounded with respect to the size of the convection the estimates are also uniform with respect to the size of the zero order reaction term and also hold for the limit case of vanishing reaction

numerical solution of the nonlinear klein gordon equation using radial basis functions the nonlinear klein gordon equation is used to model many nonlinear phenomena in this paper we propose a numerical scheme to solve the one dimensional nonlinear klein gordon equation with quadratic and cubic nonlinearity our scheme uses the collocation points and approximates the solution using thin plate splines tps radial basis functions rbf the implementation of the method is simple as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme