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

the collocation method for first kind boundary integral equations on polygonal regions in this paper the collocation method for first kind boundary integral equations by using piecewise constant trial functions with uniform mesh is shown to be equivalent to a projection method for second kind fredholm equations in a certain sense this projection is an interpolation projection by introducing this technique of analysis we particularly consider the case of polygonal boundaries we give asymptotic error estimates in l 2 norm on the boundaries and some superconvergence results for the single layer potential

isogeometric boundary element analysis using unstructured t splines we couple collocated isogeometric boundary element methods and unstructured analysis suitable t spline surfaces for linear elastostatic problems we extend the definition of analysis suitable t splines to encompass unstructured control grids unstructured meshes and develop basis functions which are smooth rational polynomials defined in terms of the bezier extraction framework and which pass standard patch tests we then develop a collocation procedure which correctly accounts for sharp edges and corners extraordinary points and t junctions this approach is applied to several three dimensional problems including a real world t spline model of a propeller we believe this work clearly illustrates the power of combining new analysis suitable computer aided design technologies with established analysis methodologies in this case the boundary element method

discrete collocation methods for second kind fredholm integral equations the numerical solution of second kind fredholm integral equations by the discrete collocation method and a discrete iterated collocation method is studied orders of convergence for the two discrete methods are given for the case in which an interpolatory quadrature rule of appropriate precision is used if the kernel is sufficiently smooth these orders of convergence will be the same as those of the exact methods some numerical results are given

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

the discrete galerkin method for integral equations a general theory is given for discretized versions of the galerkin method for solving fredholm integral equations of the second kind the discretized galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the galerkin method the theoretical framework that is given parallels that of the regular galerkin method including the error analysis of the superconvergence of the iterated galerkin and discrete galerkin solutions in some cases the iterated discrete galerkin solution is shown to coincide with the nystrom solution with the same numerical integration method the paper concludes with applications to finite element galerkin methods

high order convergence for collocation of second kind boundary integral equations on polygons we propose collocation methods with smoothest splines to solve the integral equation of the second kind on a plane polygon they are based on the bijectivity of the double layer potential between spaces of sobolev type with arbitrary high regularity and involving the singular functions generated by the corners if splines of order 2m 1 are used we get quasi optimal estimates in h m norm and optimal order convergence for the h k norm if 0 le k le m numerical experiments are presented

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

first integrals integrating factors and λ symmetries of second order differential equations for a given second order ordinary differential equation ode several relationships among first integrals integrating factors and symmetries are studied the knowledge of a symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first order linear systems of partial differential equations if two nonequivalent symmetries of the equation are known then an algorithm to find two functionally independent first integrals is provided these methods include and complete other methods to find integrating factors or first integrals that are based on variational derivatives or in the prelle singer method these results are applied to several odes that appear in the study of relevant equations of mathematical physics

λ symmetries for discrete equations following the usual definition of symmetries of differential equations we introduce the analogous concept for difference equations and apply it to some examples

local error estimates for some petrov galerkin methods applied to strongly elliptic equations on curves on etudie les proprietes de convergence locales de quelques methodes de petrov galerkin appliquees aux equations fortement elliptiques sur des courbes regulieres du plan on etablit des estimations d erreurs locales pour une grande echelle de normes de sobolev

a piecewise constant collocation method using cosine mesh grading for symm s equation here we study the piecewise constant collocation method using mesh grading to solve symm s integral equation on 1 1 we give a mesh grading for which this method achieves the optimal order of convergence even though the piecewise constant galerkin method with the same mesh grading does not some numerical results are given

numerical methods for integral equations of mellin type we present a survey of numerical methods based on piecewise polynomial approximation for integral equations of mellin type including examples arising in boundary integral methods for partial differential equations on polygonal domains

time discretization of parabolic boundary integral equations in time dependent boundary integral equations a boundary element method in space can be coupled with a different type of discretization in time for the latter a procedure based on linear multistep methods is proposed which is applicable whenever the laplace transform of the fundamental solution is known the stability properties of the method are obtained from those of the underlying multistep method in the absence of a space discretization the numerical solution given by the proposed method is identical to that of a semi discretization in time of the partial differential equation by the underlying multistep method the theory is presented for the single layer potential equation of the heat equation convergence estimates which are pointwise in time and expressed in terms of the boundary data are obtained for full discretizations using galerkin or collocation boundary element methods in space numerical examples are included

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

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

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

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

the computation of conical diffraction coefficients in high frequency acoustic wave scattering when a high frequency acoustic or electromagnetic wave is scattered by a surface with a conical point the component of the asymptotics of the scattered wave corresponding to diffraction by the conical point can be represented as an asymptotic expansion valid as the wave number k rightarrow infty the diffraction coefficient is the coefficient of the principal term in this expansion and is of fundamental interest in high frequency scattering it can be computed by solving a family of homogeneous boundary value problems for the laplace beltrami helmholtz equation parametrized by a complex wave number like parameter nu on a portion of the unit sphere bounded by a simple closed contour ell and then integrating the resulting solutions with respect to nu in this paper we give the numerical analysis of a method for carrying out this computation in the case of acoustic waves via the boundary integral method applied on ell emphasizing the practically important case when the conical scat

a method of calculation of lifting flows around 2 dimensional corner shaped bodies introduction in this paper we study the flow problem of an incompressible inviscible fluid around a two dimensional corner shaped body generally one uses the potential function of such a flow which is the solution of an exterior neumann problem considering this solution as a single layer potential one then gets a fredholm integral equation of the second kind on the boundary which can be solved by collocation methods we shall use another approach let q represent the interior region of the body q the exterior one and r its boundary then the stream function of the perturbation flow i is the solution of the exterior dirichlet problem

on the convergence of collocation methods for symm s integral equation on open curves on etablit la convergence des methodes de collocation pour l equation integrale de symm sur des courbes ouvertes on confirme experimentalement les resultats theoriques sur les vitesses de convergence asymptotiques

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

variational c symmetries and euler lagrange equations abstract a generalization of the concept of variational symmetry based on prolongations allows us to construct new methods of reduction for euler lagrange equations an adapted formulation of the noether s theorem for the new class of symmetries is presented some examples illustrate how the method works in practice

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

on the relation between standard and μ symmetries for pdes we give a geometrical interpretation of the notion of prolongations of vector fields and of the related concept of symmetry for partial differential equations extending to pdes the notion of symmetry for odes we give in particular a result concerning the relationship between symmetries and standard exact symmetries the notion is also extended to the case of conditional and partial symmetries and we analyse the relation between local symmetries and nonlocal standard symmetries

high order methods for linear functionals of solutions of second kind integral equations the authors show that linear functionals e g inner products or point functionals of solutions of second kind fredholm integral equations may be found within o 1 n 2r by very simple computational schemes using the nystrom or quadrature method based on piecewise polynomials of order r on an n 1 point mesh our theory covers a class of nonsmooth wiener hopf type equations and applications include the calculation of crack energy and stress intensity factors in certain linear elastic fracture problems and the use of the boundary integral method to compute various superconvergent solutions to boundary value problems in polygonal domains a proof of the uniform convergence of nystrom s method for certain boundary integral equations on polygonal domains is also included

on the boundary element method for some nonlinear boundary value problems here we analyse the boundary element galerkin method for two dimensional nonlinear boundary value problems governed by the laplacian in an interior or exterior domain and by highly nonlinear boundary conditions the underlying boundary integral operator here can be decomposed into the sum of a monotoneous hammerstein operator and a compact mapping we show stability and convergence by using leray schauder fixed point arguments due to petryshyn and necas

spline collocation for convolutional parabolic boundary integral equations summary we consider spline collocation methods for a class of parabolic pseudodifferential operators we show optimal order convergence results in a large scale of anisotropic sobolev spaces the results cover for example the case of the single layer heat operator equation when the spatial domain is a disc

on the asymptotic convergence of spline collocation methods for partial differential equations we examine the asymptotic accuracy of the method of collocation for the approximate solution of linear elliptic partial differential equations specifically we consider the nodal collocation of a second order equation in the plane with biperiodicity conditions using tensor product smooth splines of odd degree as trial functions we prove optimal rates of convergence in l 2 for partial derivatives of the approximate solution which are of order at least two in one variable while the solution itself and its gradient converge in l 2 at rates less than the optimal approximation theoretic results

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

improved accuracy by adapted mesh refinements in the finite element method for appropriately adapted mesh refinements optimal order error estimates are proved for the finite element approximate solution of the neumann problem for the second order elliptic equation lu 8 where 8 is the dirac distribution

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

a boundary element method for a two dimensional interface problem in electromagnetics this paper presents galerkin approximations for solutions of two dimensional interface problems by solving corresponding boundary integral equations these are obtained by simple layer potential operators only due to the strong ellipticity of the integral equations the galerkin procedure converges with optimal order smoothness of the given data implies high convergence rates for the layers

deformation of lie derivative and μ symmetries we introduce in the spirit of witten s gauging of exterior differential a deformed lie derivative that allows a geometrical interpretation of and symmetries in complete analogy with standard symmetries the case of variational symmetries both for odes and for pdes is also considered in this approach leading to and conservation laws

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

stability of a spline collocation method for strongly elliptic multidimensional singular integral equations we consider a spline collocation method for strongly elliptic zero order pseudodifferential equationsp gw au f on a cube 0 1 m utilizing multilinear spline functions which are zero at the boundary we collocate at the meshpoints inside for classical strongly elliptic translation invariant pseudodifferential operators we verify the stability of the considered collocation method inl 2 afterwards form 2 and a right hand sidef h 8 s m 2 we prove an asymptotic convergence estimate

the fast fourier transform and the numerical solution of one dimensional boundary integral equations here we present a fully discretized projection method with fourier series which is based on a modification of the fast fourier transform the method is applied to systems of integro differential equations with the cauchy kernel boundary integral equations from the boundary element method and more generally to certain elliptic pseudodifferential equations on closed smooth curves we use gaussian quadratures on families of equidistant partitions combined with the fast fourier transform this yields an extremely accurate and fast numerical scheme we present complete asymptotic error estimates including the quadrature errors these are quasioptimal and of exponential order for analytic data numerical experiments for a scattering problem the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

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

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

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

two clifford s theorems for strongly group graded rings abstract let a be a strongly group graded ring in this paper we obtain the main results as following 1 in a a similar result to clifford s theorem of the decomposition of induced modules from normal subgroups it corrects the result of 4 2 frobenius reciprocity for a 3 we extend the theory of stable modules in two ways one of which is the localization theory of rings and modules in some special cases we can obtain the corresponding results of 1 2 3

on polynomial collocation for second kind integral equations with fixed singularities of mellin type we consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of mellin convolution type using a stability result by junghanns and one of the authors we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and using the asymptotics of the solution we derive the rates of convergence finally we describe an algorithm to compute the stiffness matrix based on simple gaus quadratures and an alternative algorithm based on a recursion in the spirit of monegato and palamara orsi all together an almost best approximation to the solution of the integral equation can be computed with n2 log n 2 resp n2 operations where n is the dimension of the polynomial trial space

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

duality estimates for the numerical solution of integral equations we formulate and prove aubin nitsche type duality estimates for the error of general projection methods examples of applications include collocation methods and augmented galerkin methods for boundary integral equations on plane domains with corners and three dimensional screen and crack problems for some of these methods we obtain higher order error estimates in negative norms in cases where previous formulations of the duality arguments were not applicable

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

the convergence of a direct bem for the plane mixed boundary value problem of the laplacian in this paper the convergence analysis of a direct boundary elecment method for the mixed boundary value problem for laplace equation in a smooth plane domain is given the method under consideration is based on the collocation solution by constant elements of the corresponding system of boundary integral equations we prove the convergence of this method provide asymptotic error estimates for the bem solution and give some numerical examples

on spline galerkin methods for singular integral equations with piecewise continuous coefficients this article analizes the convergence of the galerkin method with polynomial splines on arbitrary meshes for systems of singular integral equations with piecewise continuous coefficients inl 2 on closed or open ljapunov curves it is proved that this method converges if and for scalar equations and equidistant partitions only if the integral operator is strongly elliptic in some generalized sense using the complete asymptotics of the solution we provide error estimates for equidistant and for special nonuni form partitions

a λ symmetry based method for the linearization and determination of first integrals of a family of second order ordinary differential equations we present a characterization of the second order ordinary differential equations odes that can be linearized by means of certain nonlocal transformations this characterization is given in terms of the coefficients of the equation and also determines the second order odes that admit symmetries and first integrals of some specific forms systematic methods to find these first integrals symmetries and the linearizing transformations are also derived

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

on the reduction methods for ordinary differential equations as tandard method based on the use of differential invariants of a lie group g e nables us to reduce any ordinary differential equation invariant under the action of g w e show that this method is applicable to vector fields more general than those associated with lie symmetries we characterize all such vector fields and study their relationship with nonlocal symmetries and symmetries govinder k s and leach p g l 1995 j phys a math gen 28 5349 59 muriel c and romero l 2001 ima j appl math 66 111 25

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

an analysis of quadrature errors in second kind boundary integral methods this paper considers the effect of quadrature errors on galerkin methods for approximating the solution of second kind fredholm integral equations with weakly singular kernels defined on a smooth closed surface in three dimensions a quadrature strategy for galerkin methods that use piecewise poly nomials both continuous and discontinuous on a quadrilateral based mesh is proposed the error resulting from this further discretization is proved to be no larger than the original discretization error and moreover the amount of work needed to compute the resulting matrix for the corresponding linear system is shown to be proportional to the number of matrix entries

a generalization of λ symmetry reduction for systems of odes σ symmetries we consider a deformation of the prolongation operation defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones this maintains the invariants by differentiation property and can hence be used to reduce odes satisfying suitable invariance conditions in a fully algorithmic way similarly to what happens for standard prolongations and symmetries

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

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

noether theorem for μ symmetries we give a version of noether theorem adapted to the framework of symmetries this extends to such case recent work by muriel romero and olver in the framework of symmetries and connects symmetries of a lagrangian to a suitably modified conservation law in some cases this conservation law actually reduces to a standard one we also note a relation between symmetries and conditional invariants we also consider the case where the variational principle is itself formulated as requiring vanishing variation under prolonged variation fields leading to modified euler lagrange equations in this setting symmetries of the lagrangian correspond to standard conservation laws as in the standard noether theorem we finally propose some applications and examples

on the geometry of lambda symmetries and pde reduction we give a geometrical characterization of prolongations of vector fields and hence of symmetries of odes this allows an extension to the case of pdes and systems of pdes in this context the central object is a horizontal 1 form and we speak of prolongations of vector fields and symmetries of pdes we show that these are as good as standard symmetries in providing symmetry reduction of pdes and systems and explicit invariant solutions

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

fast isogeometric boundary element method based on independent field approximation abstract an isogeometric boundary element method for problems in elasticity is presented which is based on an independent approximation for the geometry traction and displacement field this enables a flexible choice of refinement strategies permits an efficient evaluation of geometry related information a mixed collocation scheme which deals with discontinuous tractions along non smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions all these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations the system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis for the required geometrical bisection of the domain a strategy for the evaluation of bounding boxes containing the supports of nurbs basis functions is presented the versatility and accuracy of the proposed methodology are demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions

a group theoretic approach to a class of second order ordinary differential equations not possessing lie point symmetries we consider the solution of a class of second order ordinary differential equations not possessing lie point symmetries by group theoretic means the method involves increasing the order of these equations using homogeneity symmetry the solution of this new third order equation is then sought in the instances that it and the new reduced second order equation possess additional symmetries as a result the number of second order equations solvable by the lie theory of extended groups is increased

spline collocation for a boundary integral equation on polygons with cuts in this paper we construct collocation methods for a modified integral equation of the second kind on the boundary of a polygon with a finite number of cuts cuts are often met in practice and are not treated by the classical methods the double layer potential which degenerates in this situation is modified by the addition of two kinds of terms on each cut they take into account the highly singular functions generated by the cut and lead to a bijective boundary operator this operator remains bijective between spaces of sobolev types with high orders of regularity this allows the construction of collocation methods with quasi optimal estimates and high order of convergence with respect to strong norms they use smoothest splines of odd degree numerical examples of the methods are presented

local and nonlocal solvable structures in the reduction of odes solvable structures likewise solvable algebras of local symmetries can be used to integrate scalar ordinary differential equations odes by quadratures solvable structures however are particularly suitable for the integration of odes with a lack of local symmetries in fact under regularity assumptions any given ode always admits solvable structures even though finding them in general could be a very difficult task in practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra in this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order i e classical and higher in particular we introduce the notion of nonlocal solvable structure

the convergence of nystrom methods for wiener hopf equations we consider second kind integral equations on the half line where the integral operator is a compact perturbation of a convolution operator it is shown that these may be solved numerically by nystrom methods based on composite quadrature rules provided the underlying mesh is graded to correctly match the behaviour of the solution we prove the same rates of convergence that occur when the methods are applied to equations on finite intervals numerical examples are given

on the convergence of collocation methods for boundary integral equations on polygons the integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions convergence proofs and error estimates for these approximation methods have been only available in the following cases fredholm integral equations of the second kind 4 7 one dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves 2 3 17 26 29 and some special results on the classical neumann integral equation of potential theory for polygonal plane domains 5 8 9 here we give convergence proofs for collocation with piecewise linear trial functions for neumann s integral equation and symm s integral equation on plane curves with corners we derive asymptotic error estimates in sobolev norms and analyze the effect of graded meshes 0 introduction in this paper we give convergence proofs and asymptotic error estimates in sobolev norms for collocation with piecewise linear spline trial func tions applied to two basic integral equations of potential theory on plane polygons namely the integral equation of the second kind with the double layer potential neumann s integral equation and the integral equation of the first kind with the simple layer potential symm s integral equation we use an idea of arnold and wendland 2 namely considering dirac delta functions the test functions in the collocation method as second derivatives of piecewise linear functions therefore similar results as presented here should be possible for splines of higher odd order corresponding results for even order splines are not yet available thus for one of the simplest methods of numerically solving dirichlet s problem on a plane domain with corners the midpoint collocation with piecewise constant trial functions for the first kind integral equation with the simple layer potential convergence is still an open problem the method of fourier series that yields the convergence proof in the case of a smooth boundary 27 cannot be applied in the presence of corners we apply the method of local mellin transformation that has previously been used to derive error estimates for galerkin methods for a wide class of operators including those occurring in boundary element methods in acoustics electromag

nonlocal aspects of λ symmetries and odes reduction a reduction method of odes not possessing lie point symmetries makes use of the so called symmetries muriel and romero 2001 ima j appl math 66 111 the notion of covering for an ode is used here to recover symmetries of as nonlocal symmetries in this framework by embedding into a suitable system determined by the function any symmetry of can be recovered by a local symmetry of as a consequence the reduction method of muriel and romero follows from the standard method of reduction by differential invariants applied to

a modified nystrom method for integral equations with mellin type kernels the aim of this paper is to propose a new modified nystrom method for the approximation of the solutions of second kind integral equations with fixed singularities of mellin convolution type the stability and the convergence are proved in l 2 spaces and error estimates in l 2 norm are given finally numerical tests showing the effectiveness of the method are 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