a linear domain decomposition method for partially saturated flow in porous media abstract the richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated unsaturated flow in porous media we assume that the medium occupies a bounded lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface this leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at after an euler implicit discretisation of the resulting nonlinear subproblems a linear iterative l type domain decomposition scheme is proposed the convergence of the scheme is proved rigorously in the last part we present numerical results that are in line with the theoretical finding in particular the convergence of the scheme under mild restrictions on the time step size we further compare the scheme to other approaches not making use of a domain decomposition namely we compare to a newton and a picard scheme we show that the proposed scheme is more stable than the newton scheme while remaining comparable in computational time even if no parallelisation is being adopted after presenting a parametric study that can be used to optimise the proposed scheme we briefly discuss the effect of parallelisation and give an example of a four domain implementation

numerical analysis of newton projection methods for nonlinear singular integral equations abstract numerical methods of newton iteration type for the solution of two classes of nonlinear cauchy singular integral equations are studied the existence of appropriate initial approximations the solvability of the linearized discrete equations and the weighted l2 convergence of the solutions of the discrete equations to a solution of the original problem are shown

approximations of singular integral equations on lyapunov contours in banach spaces a general approximation method is investigated for the numerical evaluation of the singular integral equations on lyapunov contours defined in banach spaces the method consists in the application of the faber polynomials and the faber laurent expansion first some theorems are proved for the approximation of functions in a complex domain while these are defined in the banach space h c c 0 c 1 where c denotes a closed lyapunov contour these results are further used in order to prove the existence and uniqueness of the solutions for the systems on which the singular integral equations are reduced

on the stability of the classical collocation method for cauchy type singular integral equations abstract the classical collocation method for cauchy type singular integral equations of the second kind on an interval using gaussian quadrature rules with respect to jacobi nodes is investigated in general the nodes and the weights of the quadrature rules can only be evaluated approximately in the present paper the question is answered how precise our knowledge about them must be in order to ensure that the corresponding algebraic system of equations is solvable yet and its solution is close to the solution of the exact system

the discrete sloan iterate for the generalized airfoil equation abstract we have defined and established the superconvergence of the sloan iterate obtained from a polynomial galerkin approximation for solving the generalized airfoil equation when all the integrals involved are evaluated by using gaussian quadratures with the same number of nodes as basis elements the resulting discrete sloan iterate becomes the nystrom approximation obtained by using the same rules from this it follows that the discrete sloan iterate v n and the discrete galerkin approximation v n agree at the quadrature nodes thus v n converges no faster locally than v n for practical purposes computational superconvergence does not occur

the perturbed galerkin method for cauchy singular integral equations with constant coefficients we have considered the numerical solution of cauchy singular integral equations with constant coefficients via a polynomial based galerkin method l 2 and uniform convergence results are proved under holder continuity conditions on the kernel and right hand side and taking into account the quadrature errors produced when inner products are replaced by numerical quadratures these results generalize those of miel who considered only the case of equations of the first kind with index 0

stability and convergence of finite element approximation schemes for harmonic maps this article discusses stability and convergence of approximation schemes for harmonic maps a finite element discretization of an iterative algorithm due to f alouges is introduced and shown to be stable and convergent in general only on acute type triangulations an break a posteriori criterion is proposed which allows us to monitor sufficient conditions for weak convergence to a harmonic map on general triangulations and for adaptive mesh refinement numerical break experiments show that an adaptive strategy automatically refines triangulations in neighborhoods of typical point singularities and thereby underline its efficiency

on the numerical resolution of the generalized airfoil equation with possio kernel in this paper we consider the so called generalized airfoil singular integral equation with a smooth input function and present ad hoc quadrature rules to compute efficiently the elements of collocation and galerkin matrices

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

on long time evolution in general relativity and geometrization of 3 manifolds this paper introduces relations between the long time asymptotic behavior of einstein vacuum space times and the geometrization of 3 manifolds envisioned by thurston the relations are obtained by analysing the asymptotic behavior of a cmc foliation by compact cauchy surfaces and the induced curve of 3 manifold geometries the cheeger gromov theory is introduced in this context and a number of open problems are considered from this viewpoint

sinc nystro m method for numerical solution of a dominant system of cauchy singular integral equations given on a piecewise smooth contour a numerical method is proposed for obtaining an approximate solution to a dominant system of cauchy singular integral equations dscsies taken over a piecewise smooth contour l consisting of a finite number of arcs l j first the dscsies are reduced to a system of fredholm integral equations of the second kind which are then solved approximately by nystrom s method based on a sinc quadrature rule it is shown that the error of an n point approximation converges to zero at the rate o e cn 1 2 as n to infty provided that the coefficients of the dscsies are analytic in neighborhoods d j of each arc l j and satisfy a lipschitz condition in each d j

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

invariant formulation for the minimum induced drag conditions of nonplanar wing systems under the hypotheses of linear potential flow and rigid wake aligned with the freestream a configuration invariant analytical formulation for the induced drag minimization of single wing nonplanar systems is presented following a variational approach the resulting euler lagrange integral equation in the unknown circulation distribution is obtained the kernel presents a singularity of the first order and an efficient computational method ideal for the early conceptual phases of the design is proposed munk s theorem on the normalwash and its relation with the geometry of the wing under optimal conditions is naturally obtained with the present method moreover munk s constant of proportionality not provided in his original work is demonstrated to be the ratio between the freestream velocity and the optimal aerodynamic efficiency the augmented munk s minimum induced drag theorem is then formulated additional induced drag theorems are demonstrated following the derivations of this invariant proced

optimized schwarz methods with robin transmission conditions for parabolic problems in this paper optimized schwarz methods with robin transmission conditions are considered for solving the time dependent linear algebraic systems resulting from finite element approximations for parabolic problems we use both analysis and numerical experiments to investigate the convergence behavior particularly we check the influence of the time step size on the convergence rate we get the estimate of the upper bound of convergence rate 1 o min h 1 2 h 1 2 h 1 2 h 3 2 tau 1 where h is the mesh size h is the size of subdomains and tau is the time step size our numerical results show that the convergence rate of this method is fast we also search for the optimal parameter lambda by experiments and find the rule of its distribution

numerical analysis for one dimensional cauchy singular integral equations the paper presents a selection of modern results concerning the numerical analysis of one dimensional cauchy singular integral equations in particular the stability of operator sequences associated with different projection methods the aim of the paper is to show the main ideas and approaches such as the concept of transforming the question of the stability of an operator sequence into an invertibility problem in a certain banach algebra or the concept of certain scales of weighted besov spaces to prove convergence rates of the sequence of the approximate solutions moreover computational aspects in particular the construction of fast algorithms are discussed

a general framework for solving riemann hilbert problems numerically a new numerical framework for the approximation of solutions to matrix valued riemann hilbert problems is developed based on a recent method for the homogeneous painleve ii riemann hilbert problem we demonstrate its effectiveness by computing solutions to other painleve transcendents an implementation in mathematica is made available online

a parallel robin robin domain decomposition method for the stokes darcy system we propose a new parallel robin robin domain decomposition method for the coupled stokes darcy system with beavers joseph saffman jones interface boundary condition in particular we prove that with an appropriate choice of parameters the scheme converges geometrically independent of the mesh size

a robin type domain decomposition method with red black partition in this paper we present a new robin type nonoverlapping domain decomposition dd preconditioner the unknown variables to be solved in this preconditioned algebraic system are the robin transmission data on the interface which are different from the well known dd methods like substructuring nonoverlapping dd method and feti method through choosing suitable parameter on each subdomain boundary and using the tool of energy estimate for the second order elliptic problem we prove that the condition number of the preconditioned system is c 1 log frac h h 2 where h is the coarse mesh size and h is the fine mesh size moreover in this paper we always assume that there is a red black partition for the whole domain omega numerical results are given to illustrate the efficiency of our dd preconditioner

a remark on the application of interpolatory quadrature rules to the numerical solution of singular integral equations abstract a cauchy type singular integral equation of the first or the second kind can be numerically solved either directly or after its reduction by the usual regularization procedure to an equivalent fredholm integral equation of the second kind the equivalence of these two methods that is the equivalence both of the systems of linear algebraic equations to which the singular integral equation is reduced and of the natural interpolation formulae is proved in this paper for a class of cauchy type singular integral equations of the first kind and of the second kind but with constant coefficients for general interpolatory quadrature rules under sufficiently mild assumptions the present results constitute an extension of a series of previous results concerning only gaussian quadrature rules based on the corresponding orthogonal polynomials and their properties

on the breakdown criterion in general relativity we give a geometric criterion for the breakdown of an einstein vacuum space time foliated by a constant mean curvature or maximal foli ation more precisely we show that the foliated space time can be extended as long as the the second fundamental form and the first derivatives of the logarithm of the lapse of the foliation remain uniformly bounded we make no restrictions on the size of the initial data

on the collocation methods for singular integral equations with hilbert kernel in the present paper we introduce some singular integral operators singular quadrature operators and discretization matrices of singular integral equations with hilbert kernel these results both improve the classical theory of singular integral equations and develop the theory of singular quadrature with hilbert kernel then by using them a unified framework for various collocation methods of numerical solutions of singular integral equations with hilbert kernel is given under the framework it is very simple and obvious to obtain the coincidence theorem of collocation methods then the existence and convergence for constructing approximate solutions are also given based on the coincidence theorem

a fast and well conditioned spectral method for singular integral equations abstract we develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing chebyshev and ultraspherical polynomials to reformulate the equations as almost banded infinite dimensional systems this is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels the resulting system can be solved in o m 2 n operations using an adaptive qr factorization where m is the bandwidth and n is the optimal number of unknowns needed to resolve the true solution the complexity is reduced to o m n operations by pre caching the qr factorization when the same operator is used for multiple right hand sides stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity applications considered include the faraday cage and acoustic scattering for the helmholtz and gravity helmholtz equations including spectrally accurate numerical evaluation of the far and near field solution the julia software package singularintegralequations jl implements our method with a convenient user friendly interface

on a parallel robin type nonoverlapping domain decomposition method in recent years a nonoverlapping robin type domain decomposition method ddm for the finite element discretization systems of the second order elliptic equations which is based on using robin type boundary conditions as information transmission conditions on the subdomain interfaces has been developed and analyzed since it was first proposed by p l lions in on the schwarz alternating method iii a variant for nonoverlapping subdomains in proceedings of the 3rd international symposium on domain decomposition methods for partial differential equations siam philadelphia pa 1990 pp 202 223 however the convergence rate of this ddm with many subdomains remains open when the lower term of equations vanishes this open problem will be considered in this paper the convergence rate is almost 1 o h 1 2 h 1 2 in certain cases for example the case of a small number of subdomains where h is the mesh size and h is the size of subdomain in order to get the desirous convergence results two ma

on approximate solution of linear singular integral equations we investigate some properties of solutions of a class of linear singular integral equations the basic approach consists in replacing the singular kernel by a family of regular fredholm kernels projection techniques are used to discuss the convergence properties of solutions of these fredholm equations

cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution cauchy singular integral equations on an interval are studied in weighted spaces of continuous functions convergence rates for a general method for their numerical solution are proved in weighted uniform norms applications of these results to the collocation method and the quadrature method are given

spectral analysis of dirichlet neumann operators and optimized schwarz methods with robin transmission conditions in this paper the tight relationship between dirichlet neumann d n operators and optimized schwarz methods with robin transmission conditions is disclosed we describe the spectral distribution of continuous d n operators and give a rigorous spectral analysis of discrete d n operators by these results we prove that the optimized schwarz methods with robin transmission conditions cannot converge geometrically in the case of continuous problems furthermore we get the accurate convergence rate of the two subdomain case in addition an estimation of convergence rate of the optimized schwarz methods is presented in the general case most of our results are asymptotically sharp

the classical collocation method for singular integral equations first of all a method of collocation which we call classical collocation is described for the approximate solution of complete singular integral equations with cauchy kernel taken over the arc 1 1 secondly we demonstrate that under reasonable conditions the approximate solutions converge to the solution of the original equation

generalizations and accelerations of lions nonoverlapping domain decomposition method for linear elliptic pde in this paper a one parameter generalization of lions nonoverlapping domain decomposition method for linear elliptic pdes is proposed and studied the generalized methods are shown to be descent direction methods for minimizing an interface bias functional iteration convergence of both the continuous and finite element versions of the proposed methods is established it is theoretically and numerically demonstrated that for generic choices of the parameter the generalized methods converge faster than lions original method algorithms are given and numerical results are presented

optimized schwarz methods for model problems with continuously variable coefficients optimized schwarz methods perform better than classical schwarz methods because they use more effective transmission conditions between subdomains these transmission conditions are determined by optimizing the convergence factor which is obtained by fourier analysis for simple two subdomain model problems such optimizations have been performed for many different types of partial differential equations but almost exclusively based on the assumption of constant coefficients because only then fourier analysis can be applied we use in this paper the technique of separation of variables to study optimized schwarz methods for a model problem with a continuously variable reaction term and a similar analysis could be performed as well for many other problems with variable coefficients we obtain several new interesting results first we show that the technique of separation of variables can successfully decouple the spatial variables and give the convergence factor of subdomain iterations as a function of

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

a note on solving cauchy integral equations of the first kind by iterations abstract the successive approximation method was applied for the first time by n i ioakimidis to solve practical cases of a cauchy singular integral equation the airfoil one in this paper we study a more general case we prove the convergence of the method in this general case the proposed method has been tested for two kernels which are particularly important in practice finally some numerical examples illustrate the accuracy of the method

discrete projection methods for cauchy singular integral equations with constant coefficients we consider proving the mean square and uniform convergence of various discrete galerkin and collocation methods for cauchy singular integral equations with constant coefficients for galerkin s method the convergence rates improve those given in an earlier paper while for collocation methods the results appear to be new in particular we consider product quadrature methods for equations with discontinuous coefficients and develop and prove the convergence of a new algorithm for equations of the first kind with index zero and logarithmically singular kernels

optimized multiplicative additive and restricted additive schwarz preconditioning we demonstrate that a small modification of the multiplicative additive and restricted additive schwarz preconditioner at the algebraic level motivated by optimized schwarz methods defined at the continuous level leads to a significant reduction in the iteration count of the iterative solver numerical experiments using finite difference and spectral element discretizations of the positive definite helmholtz problem and an idealized climate simulation illustrate the effectiveness of the new approach

rates of convergence for the method of classical collocation for solving singular integral equations the method of classical collocation for the approximate solution of the complete singular integral equation with cauchy kernel taken over the arc 1 1 has been described previously by the author in siam j numer anal 19 1982 pp 816 832 in this paper we consider in greater detail the urates of convergence for both discrete and global errors in the approximate solution of this equation this analysis gives a better convergence theorem than that given in the earlier paper the paper concludes with a discussion on the necessity of a particular condition for discrete convergence

a rational function approximation of the singular eigenfunction of the monoenergetic neutron transport equation demonstrates how a proper rational fraction approximation to the singular eigenfunction of the neutron transport theory can be constructed based on the properties of generalised functions and singular integral equations the parameters of the approximant are determined by a proper use of the orthogonality integrals satisfied by the case eigenfunctions this ensures the convergence of the approximant to its exact singular distributional form use of lebesgue integrable spaces made in the analysis leads to a new possibility of approximating functions in lp 1 p infinity and also of finding approximate solutions of singular integral equations

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

finite volume methods on spheres and spherical centroidal voronoi meshes we study in this paper a finite volume approximation of linear convection diffusion equations defined on a sphere using the spherical voronoi meshes in particular the spherical centroidal voronoi meshes the high quality of spherical centroidal voronoi meshes is illustrated through both theoretical analysis and computational experiments in particular we show that the l2 error of the approximate solution is of quadratic order when the underlying mesh is given by a spherical centroidal voronoi mesh we also demonstrate numerically the high accuracy and the superconvergence of the approximate solutions

sinc nystrom method for numerical solution of one dimensional cauchy singular integral equation given on a smooth arc in the complex plane on developpe une methode numerique basee sur les fonctions sinc pour la solution approchee d une equation integrale singuliere de cauchy a une dimension sur un arc regulier du plan complexe

on the boundedness of effective potentials arising from string compactifications we study effective potentials coming from compactifications of string theory we show that under mild assumptions such potentials are bounded from below in four dimensions giving an affirmative answer to a conjecture proposed by the second author in douglas jhep 3 71 2010 we also derive some sufficient conditions for the existence of critical points all proofs and mathematical hypotheses are discussed in the context of their relevance to the physics of the problem

rational interpolation and recursive solution of lowner vandermonde systems of equations abstract rational interpolation problems for functions having numerator degree higher than denominator degree are connected with solutions of systems of equations the coefficient matrix of which has a mixed structure the first columns are of vandermonde type whereas the last columns form a lowner matrix here three term recursions for the rational interpolants are developed which can be translated into recurrence formulas for the solutions of homogeneous systems with such a coefficient matrix on this base an o n2 algorithm for the solution of n n nonhomogeneous lowner vandermonde systems is obtained

the mass of asymptotically hyperbolic manifolds motivated by certain problems in general relativity and riemannian geometry we study manifolds which are asymptotic to the hyperbolic space in a certain sense it is shown that an invariant the so called total mass can be defined unambiguously a positive mass theorem is established by using the spinor method

lower bounds for two level additive schwarz preconditioners with small overlap lower bounds for the condition numbers of the preconditioned systems are obtained for two level additive schwarz preconditioners they show that the known upper bounds for both second order and fourth order problems are sharp in the case of a small overlap

optimized schwarz methods optimized schwarz methods are a new class of schwarz methods with greatly enhanced convergence properties they converge uniformly faster than classical schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical schwarz methods if the overlap is of the order of the mesh parameter which is often the case in practical applications they achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem we analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new finite element tearing and interconnect feti variants

optimized schwarz methods for circular domain decompositions with overlap optimized schwarz methods are based on transmission conditions between subdo mains which are optimized for the problem class that is being solved such optimizations have been performed for many different types of partial differential equations but were almost exclusively based on the assumption of straight interfaces we study in this paper the influence of curvature on the optimization and we obtain four interesting new results first we show that the curvature does indeed enter the optimized parameters and the contraction factor estimates second we develop an asymptotically accurate approximation technique based on turan type inequalities in our case to solve the much harder optimization problem on the curved interface and this approximation tech nique will also be applicable to currently too complex best approximation problems in the area of optimized schwarz methods third we show that one can obtain transmission conditions from a simple circular model decomposition which have also been found using microlocal analysis but that these are not the best choices for the performance of the optimized schwarz method finally we find that in the case of curved interfaces optimized schwarz methods are not necessarily convergent for all admissible parameters our optimization leads however to parameter choices that give the same good performance for a circular decomposition as for a straight interface decomposition we illustrate our analysis with numerical experiments

superconvergence of quadratic finite elements on mildly structured grids superconvergence estimates are studied in this paper on quadratic finite element discretizations for second order elliptic boundary value problems on mildly structured triangular meshes for a large class of practically useful grids the finite element solution u h is proven to be superclose to the inter polant u i and as a result a postprocessing gradient recovery scheme for u h can be devised the analysis is based on a number of carefully derived identities in addition to its own theoretical interests the result in this paper can be used for deriving asymptotically exact a posteriori error estimators for quadratic finite element methods

rates of convergence for collocation with jacobi polynomials for the airfoil equation abstract in this paper we consider for the classical airfoil equation a collocation method based on jacobi polynomials the error is estimated in suitable nonstandard sobolev norms which are defined in such a way to respect the singularity structure of the exact solution furthermore our numerical experiments underline our optimal theoretical error estimates

a sinc hunter quadrature rule for cauchy principal value integrals a sine function approach is used to derive a new hunter type quadrature rule for the evaluation of cauchy principal value integrals of cer tain analytic functions integration over a general arc in the complex plane is considered special treatment is given to integrals over the interval 1 1 it is shown that the quadrature error is of order 0 e where n is the number of nodes used and where c is a positive constant which is indepen dent of n an application of the rule to the approximate solution of cauchy singular integral equations is also discussed numerical examples are included to illustrate the performance of the rule

finite element formulation in flat coordinate spaces to solve elliptic problems on general closed riemannian manifolds we apply finite element methods to elliptic problems on closed riemannian manifolds the elliptic equations on manifolds are reduced to coupled equations on euclidean spaces by using coordinate charts the advantage of this strategy is to avoid global triangulations on curved manifolds in the finite element method the resulting finite element problems can be solved by direct methods or by domain decomposition iterations we present a convergence theory and some computational results our methods are illustrated by a 3 dimensional sphere problem in 4 dimensional euclidean space we also give a numerical comparison between the surface finite element method and our methods

application of quadrature rules for cauchy type integrals to the generalized poincare bertrand formula the classical poincare bertrand transposition formula for the inversion of the order of integration in repeated cauchy type integrals is generalized in accordance with a new interpretation of cauchy type integrals next the gauss jacobi quadrature rule is applied in a particular case of the generalized poincare bertrand formula to both members of this formula and it is proved that this formula still remains valid after the approximation of the integrals by quadrature sums two simple applications of this result one concerning the convergence of a quadrature rule for repeated cauchy type integrals and the other the numerical solution of singular integral equations are made further generalizations and applications of the present results follow easily

a fast algorithm for solving the generalized airfoil equation abstract the present paper is devoted to the quadrature method for cauchy type singular integral equations on 1 1 with an additional logarithmic singularity of the kernel an algorithm being based on this method is given whose realization only requires a computational work of o n ln n operations in order to determine n unknown coefficients of the approximate solution furthermore in a scale of weighted sobolev like spaces on 1 1 having analogous properties like the classical sobolev spaces on the unit circle the order of convergence is optimal

convergence estimates for an higher order optimized schwarz method for domains with an arbitrary interface optimized schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations optimized schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence in the literature the analysis of optimized schwarz methods relies on fourier analysis and so the domains are restricted to be regular rectangular in this paper we express the interface operator of an optimized schwarz method in terms of poincare steklov operators this enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1 o h 1 4 on a class of general domains where h is the discretization parameter this is the predicted rate for a second order optimized schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations

a collocation method for nonlinear cauchy singular integral equations collocation methods with respect to the chebyshev nodes of first and of second kind together with both the modified and the classical newton iteration method for a class of nonlinear cauchy singular integral equations are investigated the proof of the convergence of the newton methods is based on the stability of the respective collocation methods applied to linear cauchy singular integral equations which is in turn proved by using banach algebra techniques the effective numerical realization of the methods and their applicability to a nonlinear cauchy singular integral equation arising from a free surface seepage problem are discussed

superconvergence and gradient recovery of linear finite elements for the laplace beltrami operator on general surfaces superconvergence results and several gradient recovery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the laplace beltrami equation on general surfaces with mildly structured triangular meshes for a large class of practically useful grids the surface linear finite element solution is proven to be superclose to an interpolant of the exact solution of the laplace beltrami equation and as a result various postprocessing gradient recovery including simple and weighted averaging local and global l 2 projections and zienkiewicz and zhu z z schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution numerical experiments are presented to confirm the theoretical results

the singular value decomposition of the gauss chebyshev and lobatto chebyshev methods for cauchy singular integral equations we obtain a closed form expression for the singular values and singular vectors of two matrices arising in the approximation of cauchy singular integral equations csie by the gauss chebyshev and lobatto chebyshev quadratures we also derive their singular value decomposition we use this decomposition to investigate the convergence of an iterative method for csie s proposed by ioakimidis an iterative algorithm for the numerical solution of singular integral equations j computational physics 43 164 176 1981 we show that this iterative method converges only under very restrictive conditions

analysis of schwarz methods for a hybridizable discontinuous galerkin discretization schwarz methods are attractive parallel solvers for large scale linear systems obtained when partial differential equations are discretized for hybridizable discontinuous galerkin hdg methods this is a relatively new field of research because hdg methods impose continuity across elements using a robin condition while classical schwarz solvers use dirichlet transmission conditions robin conditions are used in optimized schwarz methods to get faster convergence compared to classical schwarz methods and this even without overlap when the robin parameter is well chosen we present in this paper a rigorous convergence analysis of schwarz methods for the concrete case of the hybridizable interior penalty iph method we show that the penalization parameter needed for convergence of iph leads to slow convergence of the classical additive schwarz method and we propose a modified solver which leads to much faster convergence our analysis is entirely at the discrete level and thus holds for arbitrary in

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives