the convergence of spline collocation for strongly elliptic equations on curves most boundary element methods for two dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order the equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines the crucial assumption for the generalized boundary integral and integro differential operators is strong ellipticity our analysis is based on simple fourier expansions in particular we extend results by j saranen and w l wendland from constant to variable coefficient equations our results include the first convergence proof of midpoint collocation with piecewise constant functions i e the panel method for solving systems of cauchy singular integral equations

spline collocation for strongly elliptic equations on the torus we prove convergence and error estimates in sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus shaped domains in three or more dimensions for odd degree splines we prove convergence of nodal collocation for any strongly elliptic operator for even degree splines and midpoint collocation we find an additional condition for the convergence which is satisfied for the classical boundary integral operators our analysis is a generalization to higher dimensions of the corresponding analysis of arnold and wendland 4

numerical solution of linear volterra integral equations of the second kind with sharp gradients collocation methods are a well developed approach for the numerical solution of smooth and weakly singular volterra integral equations in this paper we extend these methods through the use of partitioned quadrature based on the qualocation framework to allow the efficient numerical solution of linear scalar volterra integral equations of the second kind with smooth kernels containing sharp gradients in this case the standard collocation methods may lose computational efficiency despite the smoothness of the kernel we illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations while keeping the solution approximation fixed the computational performance improvement introduced by our new method is examined through several test examples the final example we consider is the original problem that motivated this work the problem of calculating the probability density associated with a continuous time random walk in three dimensions that may be killed at a fixed lattice site to demonstrate how separating the solution approximation from quadrature approximation may improve computational performance we also compare our new method to several existing gregory sinc and global spectral methods where quadrature approximation and solution approximation are coupled

trigonometric collocation methods with product integration for boundary integral equations on closed curves considering boundary integral equations on smooth closed curves the product integration methods have turned out to be very efficient to define fully discrete schemes with a high convergence rate here we give a general approach to product integration methods in connection with the trigonometric collocation the discretization scheme has arbitrarily high convergence rate even exponential if the solution and the curve are analytic applications and numerical experiments confirming our theoretical results are discussed

a fredholm integral equation of the second kind for conformal mapping abstract we present a fredholm integral equation of the second kind for the derivative of the boundary correspondence function of the conformal mapping of a jordan region with piecewise twice differentiable boundary onto the unit disk one of our derivations of the equation leads to a more general regularisation procedure for some singular integral equations of the first kind the integral equation can be solved numerically by applying the fourier method the convergence to in l 2 of the solution of the discrete equation so obtained follows by simple theorems examples show that the equation often yields better numerical results than both a related method proposed by henrici and warschawski s equation

mesh grading for integral equations of the first kind with logarithmic kernel when the galerkin method is used to solve the first kind integral equations with logarithmic kernel on polygons or open arcs the singularities of the solution degrade the rates of convergence the aim of this paper is to show for the particular case of piecewise constant approximation spaces that in some cases the o h 3 order of convergence can be restored by a suitable grading of the mesh

an a posteriori error estimate for a first kind integral equation in this paper we present a new a posteriori error estimate for the boundary element method applied to an integral equation of the first kind the estimate is local and sharp for quasi uniform meshes and so improves earlier work of ours the mesh dependence of the constants is analyzed and shown to be weaker than expected from our previous work besides the galerkin boundary element method the collocation method and the qualocation method are considered a numerical example is given involving an adaptive feedback algorithm

on the numerical solution of a logarithmic integral equation of the first kind for the helmholtz equation we describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single layer approach to the dirichlet problem for the two dimensional helmholtz equation in smooth domains we develop an error analysis in a sobolev space setting and prove fast convergence rates for smooth boundary data

complementary energy principle for elastodynamics free of volumetric locking abstract this paper proposes a complementary energy principle for elastodynamic problems in doing so the stresses and the inertial forces that satisfy the equilibrium equations a priori are chosen as basic variables and then the complementary variational formulation is established by weakly enforcing the kinematic constraints the eventual formulation is posed to be an integral equation with respect to the time dimension and can be easily applied to free and forced vibration analysis in addition the volumetric locking free property inherent in the complementary energy principle is addressed numerical study on a rectangular plate is performed to validate the proposed complementary energy principle as well as to verify the featured locking free property moreover extensions of the dynamic complementary energy principle to tackle visco elastodynamics and stokes flows are also presented

a nystrom flavored calderon calculus of order three for two dimensional waves time harmonic and transient in this paper we present and test a full discretization of all elements of the calderon calculus layer potentials and integral operators for the helmholtz equation in smooth closed curves in the plane the resulting integral equations provide approximations of order three for all variables involved tests are shown for a wide array of direct indirect and combined field integral equation at fixed frequency and for a convolution quadrature based approximation in the time domain

computable upper and lower bounds on eigenfrequencies this paper is a revisit of the work ladeveze and pelle int j numer methods engrg 28 1989 where the goal here is to acquire guaranteed accurate and computable bounds of eigenfrequencies through post processing of conventional finite element results to this end a new theoretical quotient is introduced and thereafter a practical way to deal with the new quotient is developed where the constitutive relation error estimation featured with guaranteed bounding property acts as a key role academic numerical examples are performed to check the accuracy of the bounds

a highly stable and accurate computational method for eigensolutions in structural dynamics a new computational method for the linear eigensolution of structural dynamics is proposed the eigenvalue problem is theoretically transformed into a specific initial value problem of an ordinary differential equation based on the physical meaning of the sign count of the dynamic stiffness matrix a stability control device is designed and combined with the fourth order runge kutta method the resulting method finds the eigenvalues and eigenvectors at the same time with high accuracy and high stability numerical examples show that the proposed method still gives high accuracy solutions when there is a great difference in magnitude among the eigenvalues and also when some eigenvalues are very close to each other

properties of certain trigonometric series arising in numerical analysis h o f kdlsinn6 n l where 0 is real tx 0 and i 1 series of this kind have obvious connec tions with the riemann zeta function and with bernoulli polynomials and are important for fractional integration 6 xii sect 83 they have also arisen recently 2 4 in the study of certain numerical methods for bound ary integral equations on smooth curves in the plane for the applications in 2 we are interested firstly in knowing that g has a unique zero in the interval 0 rc so that a quadrature rule employing this zero is well defined and for later applications we need to know that this zero increases monotonically with c in the case where o 3 1 these questions were addressed in l as were the corresponding trickier questions about the partial sums for o b 1 we write g e p gol e b g e 4 me d fvbut fi f h e mi and for 0 p d f define k 4 b f p g 0 pi of crucial importance to the applications in 2 are the facts proved in theorem 2 below that h is nonnegative and h non positive and that k are both positive except in special cases in essence these properties ensure that the numerical method treated in 2 is stable the rate of convergence of the numerical methods of 2 is determined by the behaviour of g f i for small values of p the salient properties follow from 2 below in section 5 of the paper we indicate some extensions of our results to other classical orthogonal polynomials 2

brief modified stochastic luenberger observers a modified stochastic luenberger observer mslo structure is proposed to recover the optimal performance of the coventional slo for obtaining full state estimates in linear discrete time stochastic systems the optimal mslo omslo which gives the mmse estimates is derived by using the general two stage kalman filter a reduced order form of the omslo is also proposed for systems having singular measurement noises the connection between the omslo and the optimal minimal order observer of leondes and novak is also shown

asymptotic expansions of the error of spline galerkin boundary element methods in this paper we analyse the existence of asymptotic expansions of the error of galerkin methods with splines of arbitrary degree for the approximate solution of integral equations with logarithmic kernels these expansions are obtained in terms of an interpolation operator and are useful for the application of richardson extrapolation and for obtaining sharper error bounds we also present and analyse a family of fully discrete spline galerkin methods for the solution of the same equations following the analysis of galerkin methods we show the existence of asymptotic expansions of the error

accuracy in finite element computation for eigenfrequencies in this paper we propose an original numerical method to get upper and lower bounds for the eigenfrequencies of an elastic structure this method is based on a static formulation for eigenvalue problems built up from a new quotient rs which is defined on a load space from rs properties upper and lower bounds for the exact eigenfrequencies are proved the application of the method requires the solution of an eigenvalue problem of finite dimension and the computation of a constant which is characteristic of the discretization subspace results of numerical tests are given for the vibration problem of an elastic clamped membrane

approximation methods for the muskhelishvili equation on smooth curves we investigate the possibility of applying approximation methods to the famous muskhelishvili equation on a simple closed smooth curve since the corresponding integral operator is not invertible the initial equation has to be corrected in a special way it is shown that the spline galerkin spline collocation and spline qualocation methods for the corrected equation are stable and the corresponding approximate solutions converge to an exact solution of the muskhelishvili equation in appropriate norms numerical experiments confirm the effectiveness of the proposed methods

corrected collocation methods for periodic pseudodifferential equations we propose an approximation method for periodic pseudodifferential equations which yields higher convergence rates in sobolev spaces with negative order than the collocation method the main idea consists in correcting the usual collocation solution in a certain way by the solution of a small galerkin system for the same equation both trigonometric and spline approximation methods are considered in most of the cases our convergence result even improves that of the qualocation method

the convergence of even degree spline collocation solution for potential problems in smooth domains of the plane in this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions it turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed 1 some numerical examples which conform the theoretical results are presented

a proof of the minimal order observer in 8 it is conjectured that the minimal order observer is p n r 1 for a system with r 1 noise free measurements r 2 noisy measurements and r r 1 r 2 a proof of the existence of the minimal order observer with p n r 1 is given in this note furthermore it is shown that from this minimal order observer one can derive the familiar luenberger observer and kalman filter as two special cases

on the convergence of boundary element methods for initial neumann problems for the heat equation in this paper we study boundary element methods for initial neumann problems for the heat equation error estimates for some fully discrete methods are established numerical examples are presented

self regularization by projection for noisy pseudodifferential equations of negative order it is well known that pseudodifferential equations of negative order considered in sobolev spaces with small smoothness indices are ill posed on the other hand it is known that efficient discretization schemes with properly chosen discretization parameters allow to obtain a regularization effect for such equations the main accomplishment of the present paper is the principle for the adaptive choice of the discretization parameters directly from noisy discrete data we argue that the combination of this principle with wavelet based matrix compression techniques leads to algorithms which are order optimal in the sense of complexity

adaptive eigenfrequency analysis by superconvergent patch recovery an adaptive technique of the h version for vibration problems utilizing a new error indicator and mesh refinement strategy is proposed it is based on improvement of the eigenmodes by superconvergent patch recovery a modification of the termination criterion of the adaptive strategy is proposed the error indicator is tested by a numerical example and discussed the adaptive technique is tested by some engineering examples

a numerical scheme for solving a class of logarithmic integral equations arisen from two dimensional helmholtz equations using local thin plate splines abstract this paper presents a numerical method for solving logarithmic fredholm integral equations which occur as a reformulation of two dimensional helmholtz equations over the unit circle with the robin boundary conditions the method approximates the solution utilizing the discrete collocation method based on the locally supported thin plate splines as a type of free shape parameter radial basis functions the local thin plate splines establish an efficient and stable technique to estimate an unknown function by a small set of nodes instead of all points over the solution domain to compute logarithm like singular integrals appeared in the method we use a particular nonuniform gauss legendre quadrature rule since the scheme does not require any mesh generations on the domain it can be identified as a meshless method the error estimate of the proposed method is presented numerical results are included to show the validity and efficiency of the new technique these results also confirm that the proposed method uses much less computer memory in comparison with the method established on the globally supported thin plate splines moreover it seems that the algorithm of the presented approach is attractive and easy to implement on computers

fast solvers of integral and pseudodifferential equations on closed curves on the basis of a fully discrete trigonometric galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order u n u c n u sobolev norms of periodic functions in o n log n arithmetical operations

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

a qualocation method for burgers equation in this paper a qualocation method for the one dimensional burgers equation is proposed a semidiscrete scheme along with optimal error estimates is discussed results of a numerical experiment performed support the theoretical results

a collocation method for high frequency scattering by convex polygons we consider the problem of scattering of a time harmonic acoustic incident plane wave by a sound soft convex polygon for standard boundary or finite element methods with a piecewise polynomial approximation space the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave recently chandler wilde and langdon proposed a novel galerkin boundary element method for this problem for which by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency here we propose a related collocation method using the same approximation space for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the galerkin scheme but with a substantially reduced computational cost

projection methods and condition numbers in uniform norm for fredholm and cauchy singular integral equations in this paper the authors propose a numerical method for the approximate solution of some classes of fredholm and cauchy integral equations including the discrete collocation and collocation methods

approximation methods for singular integral equations with conjugation on curves with corners approximation methods for singular integral operators with continuous coefficients and conjugation as well as for the double layer potential operator on curves with corners are investigated with respect to their stability the methods under consideration include several kinds of quadrature rules and collocation and qualocation methods the approach is based on the thorough use of banach algebra techniques and local principles complete necessary and sufficient stability conditions are derived

fully discrete galerkin methods for systems of boundary integral equations in this paper we analyze a family of full discretizations of spline galerkin methods for a class of systems of boundary integral equations of the first kind with logarithmic principal part we prove the existence of an asymptotic expansion of the error of the galerkin and the optimal order galerkin collocation method we finally derive asymptotic expansions for some common postprocessings of the solutions both exactly and under the effect of additional discretization some examples where these techniques apply are provided

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

spline qualocation methods for variable coefficient elliptic equations on curves here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential equations of strongly and of oddly elliptic type the analysis exploits localization in the form of frozen coefficients pseudohomogeneous asymptotic symbol representation of classical pseudodifferential operators a decisive commutator property of the qualocation projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical integration of smooth remainders numerical examples show the predicted high orders of convergence

the numerical solution of symm s equation on smooth open arcs by spline galerkin methods abstract we consider the classical fredholm linear integral equation of the first kind with logarithmic kernel on a smooth jordan open arc applying the well known cosine change of variable the arc is reparametrized and the problem is transformed into a new integral equation we investigate the existence of an asymptotic expansion for the error of the galerkin method with splines on a uniform mesh as test trial functions we also analyse a full discretization of the method based on the galerkin collocation method using high order integration formulae to keep the optimal error estimates of the galerkin method in weak norms asymptotic expansions of the error for this method are provided finally we show how these expansions extend to the computation of the potential the expansions of the error in powers of the discretization parameter are useful to obtain a posteriori estimates of the error and to apply richardson extrapolation for acceleration of convergence 1999 elsevier science ltd all rights reserved

a fully discrete calderon calculus for the two dimensional elastic wave equation in this paper we present a full discretization of the layer potentials and boundary integral operators for the elastic wave equation on a parametrizable smooth closed curve in the plane the method can be understood as a non conforming petrov galerkin discretization with a very precise choice of testing functions by symmetrically combining elements on two staggered grids and using a look around quadrature formula unlike in the acoustic counterpart of this work the kernel of the elastic double layer operator includes a periodic hilbert transform that requires a particular choice of the mixing parameters we give mathematical justification of this fact finally we test the method on some frequency domain and time domain problems and demonstrate its applicability on smooth open arcs

on some hp adaptive finite element method for natural vibrations model eigenproblem results obtained by some hp adaptive fem are presented the fem is based on the texas 3 step strategy and the equilibrated residual method the focus is on numerical experiments the practical issues related to the above mentioned techniques are raised the potentials of the method for engineering applications are demonstrated

design of a state estimator for a class of time varying multivariable systems in this note matrix operators are first introduced to develop a canonical transformation then it shows that a full order state estimator with any eigenvalues can be constructed for linear time varying systems that are uniformly observable and lexicography fixed a simple algorithm is provided to design the state estimator

qualocation methods for elliptic boundary integral equations the qualocation methods developed in this paper with spline trial and test spaces are suitable for classes of boundary integral equations with convolutional principal part on smooth closed curves in the plane some of the methods are suitable for all strongly elliptic equations that is for equations in which the even symbol part of the operator dominates other methods are suitable when the odd part dominates

spline qualocation methods for boundary integral equations this paper further develops the qualocation method for the solution of integral equations on smooth closed curves qualocation is a petrov galerkin method in which the outer integrals are performed numerically by special quadrature rules here we allow the use of splines as trial and test functions and prove stability and convergence results for certain qualocation rules when the underlying petrov galerkin method is stable in the common case of a first kind equation with the logarithmic kernel a typical qualocation method employs piecewise linear splines and performs the outer integrals using just two points per interval this method with appropriate choice of the quadrature points and weights has ordero h 5 convergence in a suitable negative norm numerical experiments suggest that these rates of convergence are maintained for integral equations on open intervals when graded meshes are used to cope with unbounded solutions

on the asymptotic convergence of collocation methods with spline functions of even degree we investigate the collocation of linear one dimensional strongly elliptic integro dif ferential or more generally pseudo differential equations on closed curves by even degree polynomial splines the equations are collocated at the respective midpoints subject to uniform nodal grids of the even degree s splines we prove quasioptimal and optimal order asymptotic error estimates in a scale of sobolev spaces the results apply in particular to boundary element methods used for numerical computations in engineering applications the equations considered include fredholm integral equations of the second and the first kind singular integral equations involving cauchy kernels and integro differential equations having convolutional or constant coefficient principal parts respectively the error analysis is based on an equivalence between the collocation and certain varia tional methods with different degree splines as trial and as test functions we further need to restrict our operators essentially to pseudo differential operators having convolutional prin cipal part this allows an explicit fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing babuska s stability condition based on strong ellipticity our asymptotic error estimates extend partly those obtained by d n arnold and w l wendland from the case of odd degree splines to the case of even degree splines 1 introduction in this paper we investigate the asymptotic convergence of the collocation method using even degree polynomial splines applied to strongly elliptic systems of pseudo differential equations on closed curves with convolutional prin cipal part the collocation here employs the gauss points of one point integration as collocation points which corresponds to the usual boundary element collocation in applications 5 12 17 19 26 31 34 this is in contrast to the method investigated by g schmidt in 29 where one collocates at the break points of even order splines the asymptotic convergence properties for the standard galerkin method with splines of arbitrary orders are well known 22 for the collocation method however asymptotic convergence up to now has been shown for strongly elliptic systems only in the case of odd order splines by d n arnold and w l wendland 7

quadrature methods for 2d and 3d problems in this paper we give an overview on well known stability and convergence results for simple quadrature methods based on low order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds first we explain the methods for the case of second kind equations then we discuss what is known for the analysis of pseudodifferential equations we explain why these simple methods are not recommended for integral equations over domains with dimension higher than one finally for the solution of a two dimensional singular integral equation we prove a new result on a quadrature method based on product rules

condition of boundary integral equations arising from flow computations abstract stating flow boundary conditions for normal and tangential velocities leads to fredholm integral equations of the first and second kind approximating the vortex distribution on pressure and suction side by piecewise linear functions with equidistant knots collocation yields a commonly overdetermined system of linear equations with the vortex densities as unknowns there are several possibilities for collocation the question arises what kind of integral equation and what collocation points are to be preferred our main conclusions based on condition numbers of the resulting linear systems are the equation of the first kind with cauchy singular kernel does not appear to be worse conditioned than the equation of the second kind with regular kernel collocation points for the equation of the first kind are better placed in between the knots for the equation of the second kind at the knots the circulation is mainly determined by the equation of the first kind the mean velocity level by the equation of the second kind employing simultaneously equations of the first and second kind generally improves the solution and allows to cope with aerodynamic profiles of any thickness and loading

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

high order collocation and quadrature methods for some logarithmic kernel integral equations on open arcs this paper is devoted to the solution of the dirichlet problem for the laplace and helmholtz equation in the complement of a smooth open curve in the plane the solution is looked for as a single layer potential the corresponding density being therefore the solution of an integral equation on the open arc this equation is transformed into an equivalent 1 periodic integral equation having existence and uniqueness of solution for any periodic data here we study the use of collocation and quadrature methods for solving this equation we show the convergence of both methods and prove the existence of an asymptotic expansion of the error in powers of the discretization parameter as a consequence we show that for some of them a superconvergence phenomenon occurs when computing the solution of the differential problem two numerical experiments are shown in order to illustrate the theoretical results introduced in this work

extrapolation methods for spline collocation solutions of pseudodifferential equations on curves in this article we consider extrapolation methods in approximating smooth linear functionals by means of collocation solutions of boundary integral equations we are able to derive an asymptotic expansion of the error in main applications this expansion covers arbitrary high powers of the discretization parameter if the boundary solution is smooth the expansion gives rise to richardson type extrapolation schemes which require only simple postprocessing of calculated numbers the results are applicable for example in calculating pointwise values in the space domain when solving boundary value problems by means of integral equations in such examples the extrapolation rapidly improves the original rate of the convergence numerical experiments conform our results

a discrete collocation method for symm s integral equation on curves with corners this paper is devoted to the approximate solution of the classical first kind boundary integral equation with logarithmic kernel symm s equation on a closed polygonal boundary in 2 we propose a fully discrete method with a trial space of trigonometric polynomials combined with a trapezoidal rule approximation of the integrals before discretization the equation is transformed using a nonlinear mesh grading parametrization of the boundary curve which has the effect of smoothing out the singularities at the corners and yields fast convergence of the approximate solutions the convergence results are illustrated with some numerical examples

quadrature methods for strongly elliptic equations of negative order on smooth closed curves in this article the authors introduce and analyze a family of simple quadrature methods for strongly elliptic equations of negative order the results cover all the classical pseudodifferential equations of this kind and hence the boundary integral equations appearing in applications the quadrature methods are based on the trapezoidal approximation of the operator with respect to a uniform mesh and on varepsilon collocation they are proved to have the stability and the convergence with the maximal rate o h beta of convergence in general and o h beta 1 for some special values of varepsilon here beta is the order of the operator the method is applicable to systems of boundary integral equations as well in this connection the authors have conducted some numerical experiments which confirm their theoretical results

a spline trigonometric galerkin method and an exponentially convergent boundary integral method we consider a galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions we analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the dirichlet problem in the interior or exterior of an analytic plane curve in constrast to ordinary spline galerkin methods we show that the method is stable and so provides quasioptimal approximation in a large family of hilbert spaces including all the sobolev spaces of negative order as a consequence we prove that the approximate solution to the dirichlet problem and all its derivatives converge pointwise with exponential rate

explicit gaussian quadrature rules for c 1 cubic splines with symmetrically stretched knot sequences we provide explicit expressions for quadrature rules on the space of c 1 cubic splines with non uniform symmetrically stretched knot sequences the quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal that is it requires minimal number of nodes numerical experiments validating the theoretical results and the error estimates of the quadrature rules are also presented the paper provides expressions for quadrature rules on the space of c 1 cubic splines with non uniform symmetrically stretched knot sequences any function from the space is exactly integrated by the rule the quadrature nodes and weights are derived via explicit recursion formulae with no intervention of any numerical solver the rule is optimal i e it requires minimal number of nodes

the numerical solution of first kind logarithmic kernel integral equations on smooth open arcs consider solving the dirichlet problem au p o pem2 s u p h p pes sup u p co pes2 with s a smooth open curve in the plane we use single layer potentials to construct a solution u p this leads to the solution of equations of the form j g q og p q ds q h p pes this equation is reformulated using a special change of variable leading to a new first kind equation with a smooth solution function this new equation is split into a principal part which is explicitly invertible and a compact perturbation then a discrete galerkin method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically a complete convergence analysis is given numerical examples conclude the paper with s a smooth open contour in the plane this equation arises in a variety of contexts one being the study of elasticity crack problems in the plane we propose a new numerical method for solving 1 1 and then use it to solve some potential theory problems in the plane the method takes account of the expected singularities in g at the ends of the contour in an entirely natural way and the method is shown to converge rapidly when the curve 5 and the data are sufficiently smooth we limit the functions g and h to be real although the following development extends easily to the complex case let s have a parametrization

solution of second kind fredholm integral equations by means of gauss and anti gauss quadrature rules this paper is concerned with the numerical approximation of fredholm integral equations of the second kind a nystrom method based on the anti gauss quadrature formula is developed and investigated in terms of stability and convergence in appropriate weighted spaces the nystrom interpolants corresponding to the gauss and the anti gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation under suitable assumptions which are easily verified for a particular weight function hence an error estimate is available and the accuracy of the solution can be improved by approximating it by an averaged nystrom interpolant the effectiveness of the proposed approach is illustrated through different numerical tests

the spline collocation method for parabolic boundary integral equations on smooth curves we consider the spline collocation method for a class of parabolic pseudodifferential operators we show optimal order convergence results in a large scale of anisotropic sobolev spaces the results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators for the localization we need also some special spline approximation results in anisotropic sobolev spaces

some boundary element methods using dirac s distributions as trial functions in this article a boundary element method using dirac s distributions as trial functions is proposed as test functions we use smoother splines the computational complexity of this method is comparable with the collocation method and it converges with less regularity requirements on the boundary data asymptotic error analysis is derived an example considering solution of the potential problem is discussed in some detail finally a numerical example demonstrates the applicability of the method in practical calculations

on numerical quadrature for c1 quadratic powell sabin 6 split macro triangles abstract the quadrature rule of hammer and stroud 1956 for cubic polynomials has been shown to be exact for a larger space of functions namely the c 1 cubic clough tocher spline space over a macro triangle if and only if the split point is the barycentre of the macro triangle kosinka and barto 2018 we continue the study of quadrature rules for spline spaces over macro triangles now focusing on the case of c 1 quadratic powell sabin 6 split macro triangles we show that the 3 node gaussian quadrature s for quadratics can be generalised to the c 1 quadratic powell sabin 6 split spline space over a macro triangle for a two parameter family of inner split points not just the barycentre as in kosinka and barto 2018 the choice of the inner split point uniquely determines the positions of the edge split points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature consequently the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three for the inner split point at the barycentre we prove that the two 3 node quadratic polynomial quadratures of hammer and stroud exactly integrate also the c 1 quadratic powell sabin spline space if and only if the edge split points are at their respective edge midpoints for other positions of the inner and edge split points we provide numerical examples showing that three nodes suffice to integrate the space exactly but a full classification and a closed form solution in the generic case remain elusive

h adaptive refinement strategy for acoustic problems with a set of natural frequencies abstract the finite element method has been applied to the analysis of acoustic problems with several natural frequencies and mode shapes first a recovery based error estimation is performed following the well known procedures of structural problems then an h adaptive refinement strategy is proposed that leads to a finite element mesh with the minimum number of elements and with a specified error for each of the natural frequencies included in the analysis the procedure provides a useful numerical tool since the computational requirements are reduced in addition results obtained by means of the minimum element size procedure are shown for comparison purposes the similarity of the meshes given by the two methods is justified on the basis of the equations that lead to the element size of the mesh the procedure has been applied to some numerical examples to illustrate its validity

gaussian quadrature rules for c1 quintic splines with uniform knot vectors abstract we provide explicit quadrature rules for spaces of c 1 quintic splines with uniform knot sequences over finite domains the quadrature nodes and weights are derived via an explicit recursion that avoids numerical solvers each rule is optimal that is requires the minimal number of nodes for a given function space for each of n subintervals generically only two nodes are required which reduces the evaluation cost by 2 3 when compared to the classical gaussian quadrature for polynomials over each knot span numerical experiments show fast convergence as n grows to the two third quadrature rule of hughes et al 2010 for infinite domains

meshfree methods and boundary conditions summary we perform here some meshfree methods to inhomogeneous laplace equations we prove the efficiency of those methods compared with classical ones for one or two dimensional case for numerics and for one dimensional for theoretical results copyright 2006 john wiley sons ltd

the modified quadrature method for classical pseudodifferential equations of negative order on smooth closed curves in this article we present and analyze the modified quadrature method for strongly elliptic boundary integral equations of negative order on smooth closed curves the method employs a modification that extracts the singularity appearing in the kernel or in some of its derivatives moreover the composite trapezoidal rule is used for approximation of the integral the modified quadrature method is proved to have the maximal rate o h 1 of convergence for operators of order 0 in general and o h 2 for a large class of operators appearing in applications some numerical experiments confirming our theoretical results are also presented

a quadrature based approach to improving the collocation method the collocation method is a popular method for the approximate solution of boundary integral equations but typically does not achieve the high order of convergence reached by the galerkin method in appropriate negative norms in this paper a quadrature based method for improving upon the collocation method is proposed and developed in detail for a particular case that case involves operators with even symbol such as the logarithmic potential operating on 1 periodic functions a smooth spline trial space of odd degree with constant mesh spacingh 1 n and a quadrature rule with 2n points where ann point quadrature rule would be equivalent to the collocation method in this setting it is shown that a special quadrature rule which depends on the degree of the splines and the order of the operator can yield a maximum order of convergence two powers ofh higher than the collocation method

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis