radial discrete pde splines on lipschitz domains abstract in the framework of an elliptic partial differential equation pde certain boundary conditions and a set of points to approximate in a lipschitz domain and arbitrary dimension we use radial basis function rbf techniques for the construction and characterization of discrete pde splines we also show convergence and derive error estimates

a gradient reproducing kernel collocation method for boundary value problems summary the earlier work in the development of direct strong form collocation methods such as the reproducing kernel collocation method rkcm addressed the domain integration issue in the galerkin type meshfree method such as the reproducing kernel particle method but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of collocation points for optimal convergence in this work we intend to address the computational complexity in rkcm while achieving optimal convergence by introducing a gradient reproduction kernel approximation the proposed gradient rkcm reduces the order of differentiation to the first order for solving second order pdes with strong form collocation we also show that different from the typical strong form collocation method where a significantly large number of collocation points than the number of source points is needed for optimal convergence the same number of collocation points and source points can be used in gradient rkcm we also show that the same order of convergence rates in the primary unknown and its first order derivative is achieved owing to the imposition of gradient reproducing conditions the numerical examples are given to verify the analytical prediction copyright 2012 john wiley sons ltd

error analysis for moving least squares approximation in 2d space abstract in this paper we provide a theoretical analysis of the moving least squares mls approximation which belongs to the family of meshless methods first the non matrix form of the mls shape function in two dimensional space is obtained by using consistency conditions the error estimates for mls approximation in sobolev space are presented when u x y c m 1 and u x y w m 1 q respectively we establish the error estimates for interpolating element free galerkin iefg method when it is used for solving poisson s equation the error bound is related to the radii of the weight functions and the bound of the norm of derivatives of shape functions three numerical examples are selected to confirm our analysis

a galerkin radial basis function method for the schrodinger equation in this article we consider the discretization of the time dependent schrodinger equa tion using radial basis functions rbfs we formulate the discretized problem over an unbounded domain without imposing explicit boundary conditions since we can show that time stability of the discretization is not guaranteed for an rbf collocation method we propose to employ a galerkin ansatz instead for gaussians it is shown that exponential convergence is obtained up to a point where a systematic error from the domain where no basis functions are centered takes over the choice of the shape parameter and of the resolved region is studied numerically compared to the fourier method with periodic boundary conditions the basis functions can be centered in a smaller domain which gives increased accuracy with the same number of points

the overlapped radial basis function finite difference rbf fd method a generalization of rbf fd abstract we present a generalization of the rbf fd method that computes rbf fd weights in finite sized neighborhoods around the centers of rbf fd stencils by introducing an overlap parameter 0 1 such that 1 recovers the standard rbf fd method and 0 results in a full decoupling of stencils we provide experimental evidence to support this generalization and develop an automatic stabilization procedure based on local lebesgue functions for the stable selection of stencil weights over a wide range of values we provide an a priori estimate for the speedup of our method over rbf fd that serves as a good predictor for the true speedup we apply our method to parabolic partial differential equations with time dependent inhomogeneous boundary conditions neumann in 2d and dirichlet in 3d our results show that our method can achieve as high as a 60 speedup in 3d over existing rbf fd methods in the task of forming differentiation matrices

non matching mesh gluing by meshless interpolation an alternative to lagrange multipliers rewards of using non matching meshes are great in many aspects sub meshing a complex structure in a sub structure wise manner separately and or in parallel selective local refinement unlimited scalability guarantee for generating meshes in increasingly large scale finite element simulation etc however when meshes do not match a gluing algorithm is required to enforce inter domain continuity non matching terminologically originates from mesh discretization in nature and the non matching mesh issue is therefore avoidable or does not exist in a node based or meshless discretization in meshless methods motivated by this a gluing algorithm is developed based on meshless interpolation the gluing is accomplished by continuous trial and test functions across non matching meshes constructed using nodes only the developed gluing algorithm is tested in static and wave propagation problems in two and three dimensions it is demonstrated that the gluing algorithm does not deteriorate convergence and accuracy compared with the one piece conforming mesh compared with lagrange multiplier gluing one of the most common approaches the current algorithm has two significant benefits 1 easier implementation in any dimension and 2 positive definite banded system matrices not acting against equation solvers and hence better suited for large scale finite element analysis

overlapping radial basis function interpolants for spectrally accurate approximation of functions of eigenvalues with application to buckling of composite plates abstract many physical simulations involve eigenvalue computations natural frequencies stability engineering optimization requires many evaluations of such eigenvalues leading to excessive computational times especially for global optimization techniques accurate approximations of eigenvalues with respect to optimization variables that often are natural parameters of the physical systems are then required to alleviate the computational burden dependence of the critical eigenvalue with respect to these natural parameters is often complex part of the reason is that the critical eigenvalue is the minimum of several eigenvalues resulting in a loss of differentiability for physical parameters where the critical eigenvalue becomes multiple this discontinuous derivative prevents from accurate approximation whenever the approximation model is smooth such as most of the standard approximation techniques kriging artificial neural network in this work we present an original strategy that takes into account this discontinuous behavior by dividing the input space through the clustering of the gradient space radial basis interpolants are then constructed over each region and a patch region is defined that encompasses the non differentiable region this allows us to retrieve over the whole domain of definition excellent convergence rates and numerical experiments over realistic numerical simulation show that it is possible to achieve spectrally accurate approximations

sobolev bounds on functions with scattered zeros with applications to radial basis function surface fitting in this paper we discuss sobolev bounds on functions that vanish at scattered points in a bounded lipschitz domain that satisfies a uniform interior cone condition the sobolev spaces involved may have fractional as well as integer order we then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions rbfs these estimates include situations in which the target function does not belong to the native space of the rbf

a mixed method for dirichlet problems with radial basis functions we present a simple discretization by radial basis functions for the poisson equation with dirichlet boundary condition a lagrangian multiplier using piecewise polynomials is used to accommodate the boundary condition this simplifies previous attempts to use radial basis functions in the interior domain to approximate the solution and on the boundary to approximate the multiplier which technically requires that the mesh norm in the interior domain is significantly smaller than that on the boundary numerical experiments confirm the theoretical results

a weak form rbf generated finite difference method abstract in this paper the idea of direct discretization via radial basis functions rbfs is applied on a local petrov galerkin test space of a partial differential equation pde this results to a weak based rbf generated finite difference rbf fd scheme that possesses some useful properties the error and stability issues are considered when the pde solution or the basis function has low smoothness the new method gives more accurate results than the already well established strong based collocation methods although the method uses a galerkin formulation it still remains meshless because not only the approximation process relies on scattered point layouts but also integrations are done over non connected independent and well shaped subdomains some applications to potential and elasticity problems on scattered data points support the theoretical analysis and show the efficiency of the proposed method

a local least squares method for solving nonlinear partial differential equations of second order in this paper a mesh free method for the treatment of time independent and time dependent nonlinear pdes of second order is presented the basic idea of the discretization is a local least squares approximation similar to the moving least squares approach in data approximation however in our approach the pde is incorporated as an additional minimization constraint the discretization leads to a fixed point problem which is solved by iteration because of the local nature of the method only small dimensional matrix inversions have to be done the approximation error of the discretization even on unstructured meshes is comparable to respective versions of finite elements as a by product the method provides an a posteriori measure for the local approximation error we discuss implementational aspects and present numerical simulations

multiscale rbf collocation for solving pdes on spheres in this paper we discuss multiscale radial basis function collocation methods for solving certain elliptic partial differential equations on the unit sphere the approximate solution is constructed in a multi level fashion each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh two variants of the collocation method are considered sometimes called symmetric and unsymmetric although here both are symmetric a convergence theory is given which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions

a meshless galerkin method for dirichlet problems using radial basis functions in this paper a numerical method is given for partial differential equations which combines the use of lagrange multipliers with radial basis functions it is a new method to deal with difficulties that arise in the galerkin radial basis function approximation applied to dirichlet also mixed boundary value problems convergence analysis results are given several examples show the efficiency of the method using tps or sobolev splines

a meshless method for axi symmetric poroelastic simulations numerical study a meshless model based on the meshless local petrov galerkin mlpg approach is developed and implemented for the solution of axi symmetric poroelastic problems the solution accuracy and the code performance are investigated on a realistic application concerning the prediction of land subsidence above a deep compacting reservoir the analysis addresses several numerical issues including the parametric selection of the optimal size of the local sub domains for the weak form and the nodal supports the appropriate integration rule and the linear system solver the results show that mlpg can be more accurate than the standard finite element fe method on coarse discretizations with its superiority decreasing as the nodal resolution increases this is due to both a slower convergence rate and a progressively higher computational cost compared to fe these drawbacks can be partially mitigated by improving the efficiency of the numerical integration and the system solver with the aid of projection techniques based on krylov subspace methods the outcome of the present analysis supports the development of coupled methods where a limited number of mlpg nodes are used to locally improve a fe solution copyright 2006 john wiley sons ltd

error estimates for the interpolating moving least squares method in this paper the interpolating moving least squares imls method presented by lancaster and salkauskas is discussed in details the advantage of the imls method is that the meshless method which is constructed based on the imls method can apply the essential boundary conditions directly and easily a simpler expression of the approximation function of the imls method is obtained then the error estimate of the approximation function and its first and second order derivatives of the imls method are presented in one dimensional case in this paper the theoretical results show that if the order of the polynomial basis functions is big enough and the original function is sufficiently smooth then the approximation function and its partial derivatives are convergent to the exact values in terms of the maximum radius of the domains of influence of nodes for the purpose of demonstration some selected numerical examples are given to prove the theories in this paper

on condition number of meshless collocation method using radial basis functions in this paper we will give an estimate for the condition number of meshless collocation method using radial basis functions

a rescaled localized radial basis function interpolation on non cartesian and nonconforming grids in this paper we propose a rescaled localized radial basis function rl rbf interpolation method based on the use of compactly supported radial basis functions starting from the classical rbf interpolation technique we introduce a rescaling that allows for exact interpolation of constant fields between nonconforming meshes without the use of an extra polynomial term we also present two dimensional and three dimensional numerical examples on arbitrary finite element meshes to show that the rl rbf interpolation leads to accurate results fast evaluation and easy parallelization of the algorithm all the computations are carried out using the open source finite element library lifev

interpolation on spherical geodesic grids a comparative study most operational models in atmospheric physics meteorology and climatology nowadays adopt spherical geodesic grids and require ad hoc developed interpolation procedures the author does a comparison between chosen representatives of linear distance based and cubic interpolation schemes outlining their advantages and drawbacks in this specific application field numerical experiments on a standard test problem while confirming a good performance of linear and distance based schemes in a single interpolation step also show their minor accuracy with respect to the cubic scheme in the more realistic simulation of advection of a meteorological field

meshless galerkin method using radial basis functions based on domain decomposition in this present paper we will give an overview about the combination of rbf meshless method with domain decomposition especially galerkin method based on the analysis for this method before we will consider the combination of projection domain decomposition method with radial basis function both of galerkin approximation and collocation method are discussed some examples are given to show the efficiency of this method

aspects of nonnormality for iterative methods abstract recently new optimal krylov subspace methods have been discovered for normal matrices in light of this novel ways to quantify nonnormality are considered in connection with various families of matrices we use as a criterion how for a given matrix these iterative methods introduced can be employed via e g inexpensive matrix factorizations the unitary orbit of the set of binormal matrices provides a natural extension of normal matrices its elements yield polynomially normal matrices of moderate degree in this context several matrix nearness problems arise

local heaviside weighted mlpg meshless method for two dimensional solids using compactly supported radial basis functions abstract compactly supported radial basis functions csrbf are employed for constructing trial functions in the local heaviside weighted meshless local petrov galerkin method for stress analysis of two dimensional solids where the heaviside step function is used as the weighting function over a local sub domain the present method is a truly meshless method based only on a number of randomly located nodes no domain integration is needed no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions effects of the sizes of local sub domain and interpolation domain on the performance of the present method are investigated in this paper the size of the support of the basis function has been treated as a shape parameter and then the behaviour of this shape parameter has been systematically studied for six different csrbfs example problems in elastostatics are presented and compared with closed form solutions results show that the proposed method is highly accurate and possesses no numerical difficulties

strongly elliptic pseudodifferential equations on the sphere with radial basis functions spherical radial basis functions are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere these equations arise from geodesy the approximate solutions are found by the galerkin and collocation methods a salient feature of the paper is a unified theory for error analysis of both approximation methods

solving partial differential equations by meshless methods using radial basis functions abstract in this paper we will present some meshless methods for partial differential equations including meshless subdomain method meshless petrov galerkin method and meshless schwarz method

preconditioned conjugate gradients radial basis functions and toeplitz matrices abstract radial basis functions provide highly useful and flexible interpolants to multivariate functions further they are beginning to be used in the numerical solution of partial differential equations unfortunately their construction requires the solution of a dense linear system therefore much attention has been given to iterative methods in this paper we present a highly efficient preconditioner for the conjugate gradient solution of the interpolation equations generated by gridded data thus our method applies to the corresponding toeplitz matrices the number of iterations required to achieve a given tolerance is independent of the number of variables

operators on radial functions a general theory is provided that allows to write multi variate fourier transforms or convolutions of radial functions as very simple univariate operations as a byproduct an interesting group of operators i x r with i i i i i is defined it contains the classical derivatives as i1 d dr and is intimately connected to the fourier transform applications to the construction of new positive definite radial and to new identities for special functions are included

an explicit unconditionally stable numerical solution of the advection problem in irrotational flow fields 1 a new methodology for the eulerian numerical solution of the advection problem is proposed the methodology is based on the conservation of both the zero and the first order spatial moments inside each element of the computational domain and leads to the solution of several small systems of ordinary differential equations since the systems are solved sequentially one element after the other the method can be classified as explicit the proposed methodology has the following properties 1 it guarantees local and global mass conservation 2 it is unconditionally stable and 3 it applies second order approximation of the concentration and its fluxes inside each element limitation of the procedure to irrotational flow fields for the 2 d and 3 d cases is discussed the results of three 1 d and 2 d literature tests are compared with those obtained using other techniques a new 2 d test with radially symmetric flow is also carried out

meshless collocation method based on dirichlet neumann substructure iteration in this paper we will combine the dirichlet neumann substructure iteration with so called meshless collocation method for pde using rbf the convergence of this method and numerical examples are given to show the efficiency

an inequality for characteristic functions and its applications to uncertainty relations and the quantum zeno effect an inequality concerning characteristic functions is established it is useful in studying zero neighbourhood behaviours of characteristic functions the physical implications for the time energy uncertainty relations and the quantum zeno effect are indicated

meshless schemes for unsteady navier stokes equations in vorticity formulation using radial basis functions in this paper we provide a new scheme for unsteady incompressible flows in vorticity stream function formulation combined with the radial basis functions method it is an efficient meshless method optimal accuracy can be achieved using this method the efficiency and accuracy are demonstrated by numerical examples

reproducing kernel enhanced local radial basis collocation method standard radial basis functions rbfs offer exponential convergence however the method is suffered from the large condition numbers due to their nonlocal approximation the nonlocality of rbfs also limits their applications to small scale problems the reproducing kernel functions on the other hand provide polynomial reproducibility in a local approximation and the corresponding discrete systems exhibit relatively small condition numbers nonetheless reproducing kernel functions produce only algebraic convergence this work intends to combine the advantages of rbfs and reproducing kernel functions to yield a local approximation that is better conditioned than that of the rbfs while at the same time offers a higher rate of convergence than that of reproducing kernel functions further the locality in the proposed approximation allows its application to large scale problems error analysis of the proposed method is also provided numerical examples are given to demonstrate the improved conditioning and accuracy of the proposed method copyright 2007 john wiley sons ltd

necessary and sufficient conditions for the quantum zeno and anti zeno effect a necessary and sufficient condition for the occurrence of the quantum zeno effect is given refining a recent conjecture of luo wang and zhang an analogous condition is derived for the quantum anti zeno effect both results rely on a formal connection between the quantum anti zeno effect and the weak law of large numbers

a note on the meshless method using radial basis functions in this paper we prove some inverse inequalities for the finite dimensional subspace interpolated by radial basis functions the properties of these inequalities are worse than those in finite element spaces these bad properties result in a huge condition number for both the collocation method and galerkin method using radial basis functions as applications theoretical estimates of the condition numbers of a meshless method using radial basis functions are given

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

analysis of thick plates by using a higher order shear and normal deformable plate theory and mlpg method with radial basis functions infinitesimal deformations of a homogeneous and isotropic thick elastic plate have been analyzed by using a meshless local petrov galerkin mlpg method and a higher order shear and normal deformable plate theory honsdpt radial basis functions rbf are employed for constructing trial solutions while a spline function is used as the weight function over a local subdomain the present method uses a number of randomly distributed nodes in the domain and is truly meshless two types of rbfs i e multiquadrics mq and thin plate splines tps are employed and effects of their shape parameters on the quality of the computed solution are examined for deformations of thick plates under different boundary conditions it is found that the present mlpg formulations give results very close to those obtained by other researchers a benefit of using rbfs is that no special treatment is needed to impose the essential boundary conditions which substantially reduces the computational cost 2006 elsevier b v all rights reserved

a partition of unity based fe meshfree quad4 element with radial polynomial basis functions for static analyses abstract recently a partition of unity pu based finite element called fe meshfree quad4 element has been published that element employed polynomial basis functions for the local approximation la in the present paper a new fe meshfree quad4 element employing hybrid radial polynomial basis functions for the la is proposed an advantage of radial polynomial basis is the freedom from the possible singularity of the moment matrix that could sometimes result with an inappropriate choice of polynomial basis functions another advantage is the improved accuracy of finite element solution the new element has been applied to several linear and nonlinear test problems the results demonstrate that the new element gives much better performance as compared to the previous fe meshfree quad4 element with pure polynomial basis even with a lower order basis viz with just three or four polynomial terms the present element is capable of giving more accurate solution than the previous element which employs a six term polynomial basis the new element also exhibits a much higher degree of tolerance to mesh distortion than the known quadratic elements moreover the linear dependence problem otherwise associated with many of the partition of unity pu based elements is completely eliminated from the present element

analysis of moving least squares approximation revisited in this article the error estimation of the moving least squares approximation is provided for functions in fractional order sobolev spaces the analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details an application to galerkin method for partial differential equations is also supplied

kernel based approximation in sobolev spaces with radial basis functions in this paper we study several radial basis function approximation schemes in sobolev spaces we obtain an optional error estimate by using a class of smoothing operators we also discussed sufficient conditions for the smoothing operators to attain the desired approximation order we then construct the smoothing operators by some compactly supported radial kernels and use them to approximate sobolev space functions with optimal convergence order these kernels can be simply constructed and readily applied to practical problems the results show that the approximation power depends on the precision of the sampling instrument and the density of the available data

error estimate for the numerical solution of fractional reaction subdiffusion process based on a meshless method in this paper a numerical technique based on a meshless method is proposed for solving the time fractional reaction subdiffusion equation firstly we obtain a time discrete scheme based on a finite difference scheme then we use the meshless galerkin method to approximate the spatial derivatives and obtain a full discrete scheme in the proposed scheme some integrals appear over the boundary and the domain of problem which will be approximated using gauss legendre quadrature rule then we prove that the time discrete scheme is unconditionally stable and convergent using the energy method we show convergence order of the time discrete scheme is o the aim of this paper is to obtain an error estimate and to show convergence for the meshless galerkin method based on the radial basis functions numerical examples confirm the efficiency and accuracy of the proposed scheme

adaptive residual subsampling methods for radial basis function interpolation and collocation problems we construct a new adaptive algorithm for radial basis functions rbfs method applied to interpolation boundary value and initial boundary value problems with localized features nodes can be added and removed based on residuals evaluated at a finer point set we also adapt the shape parameters of rbfs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix the performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains

meshless galerkin methods using radial basis functions we combine the theory of radial basis functions with the field of galerkin methods to solve partial differential equations after a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions

sufficient conditions for the anti zeno effect the ideal anti zeno effect means that a perpetual observation leads to an immediate disappearance of the unstable system we present a straightforward way to derive sufficient conditions under which such a situation occurs expressed in terms of the decaying states and spectral properties of the hamiltonian they show in particular that the gap between zeno and anti zeno effects is in fact very narrow

conservative multimoment transport along characteristics for discontinuous galerkin methods a set of algorithms based on characteristic discontinuous galerkin methods is presented for tracer transport on the sphere the algorithms are designed to reduce message passing interface communica

from potential theory to matrix iterations in six steps the theory of the convergence of krylov subspace iterations for linear systems of equations conjugate gradients biconjugate gradients gmres qmr bi cgstab and so on is reviewed for a computation of this kind an estimated asymptotic convergence factor rho le 1 can be derived by solving a problem of potential theory or conformal mapping six approximations are involved in relating the actual computation to this scalar estimate these six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming

semi implicit semi lagrangian modelling for environmental problems on staggered cartesian grids with cut cells in this paper we propose an extension of semi implicit models for the shallow water or euler equations on staggered cartesian meshes to handle the presence of cut cells at the boundaries of the computational domain semi lagrangian advection algorithms are also extended to staggered cartesian meshes with cut boundary cells a linear reconstruction algorithm and radial basis function interpolation are used respectively to compute approximate trajectories and to recover an accurate velocity field in cells intersected by the boundary the effective accuracy of radial basis function interpolators for semi lagrangian algorithms is assessed by simple numerical experiments timestep control techniques in the computation of the semi lagrangian trajectories are also discussed the accuracy of the resulting models for environmental flows is demonstrated by a number of test cases simulating nonlinear open channel flows and baroclinic flows over an isolated obstacle

a quasi interpolation method for solving stiff ordinary differential equations based on the idea of quasi interpolation and radial basis functions approximation a numerical method is developed to quasi interpolate the forcing term of differential equations by using radial basis functions a highly accurate approximation for the solution can then be obtained by solving the corresponding fundamental equation and a small size system of equations related to the initial or boundary conditions this overcomes the ill conditioning problem resulting from using the radial basis functions as a global interpolant error estimation is given for a particular second order stiff differential equation with boundary layer the result of computations indicates that the method can be applied to solve very stiff problems with the use of multiquadric a special class of radial basis functions it has been shown that a reasonable choice for the optimal shape parameter is obtained by taking the same value of the shape parameter as the perturbed parameter contained in the stiff equation copyright 2000 john wiley sons ltd

generalizations of the field of values useful in the study of polynomial functions of a matrix for a given square matrix a and positive integer k we consider sets in the complex plane satisfying p a maxz p z for all polynomials p of degree k or less the largest such set referred to as the polynomial numerical hull of degree k was introduced by o nevanlinna convergence of iterations for linear equations birkhauser basel 1993 and a number of properties of this set were derived for both matrices and linear operators we give several equivalent characterizations of the polynomial numerical hull of degree k and we actually compute these sets for several matrices for k 1 this set is just the field of values of a and for k m where m is the degree of the minimal polynomial of a it is the spectrum of a for 1 k m these sets are intermediate between the field of values and the spectrum and sometimes resemble pseudospectra

radial basis collocation methods for elliptic boundary value problems this paper introduces radial basis functions rbf into the collocation methods and the combined methods for elliptic boundary value problems first the ritz galerkin method rgm is chosen using the rbf and the integration approximation leads to the collocation method of rbf for poisson s equation next the combinations of rbf with finite element method fem finite difference method fdm etc can be easily formulated by following li 1 and hu and li 2 3 but more analysis of inverse estimates is explored in this paper since the rbfs have the exponential convergence rates and since the collocation nodes may be scattered in rather arbitrary fashions in various applications the rbf may be competitive to orthogonal polynomials for smooth solutions moreover for singular solutions we may use some singular functions and rbfs together numerical examples for smooth and singularity problems are provided to display effectiveness of the methods proposed and to support the analysis made

a least squares radial basis function partition of unity method for solving pdes recently collocation based radial basis function rbf partition of unity methods pums for solving partial differential equations have been formulated and investigated numerically and theoretically when combined with stable evaluation methods such as the rbf qr method high order convergence rates can be achieved and sustained under refinement however some numerical issues remain the method is sensitive to the node layout and condition numbers increase with the refinement level here we propose a modified formulation based on least squares approximation we show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling we derive theoretical error estimates both for the collocation and least squares rbf pums numerical experiments are performed for the poisson equation in two and three space dimensions for regular and irregular geometries the convergence experiments confirm the theoretical estimates and the least squares formulation is shown to

rbf multiscale collocation for second order elliptic boundary value problems in this paper we discuss multiscale radial basis function collocation methods for solving elliptic partial differential equations on bounded domains the approximate solution is constructed in a multilevel fashion each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh on each level standard symmetric collocation is employed a convergence theory is given which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions we are able to show that the convergence is linear in the number of levels we also discuss the condition numbers of the arising systems and the effect of simple diagonal preconditioners now proving rigorously previous numerical observations

solving partial differential equations by bkm combined with ddm the boundary knot method bkm has recently been developed as an inherently meshless integration free boundary type collocation technique for the numerical discretization of general partial differential equation systems unlike the method of fundamental solutions the use of non singular radial basis functions in the bkm avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain but the condition number of the coefficient matrix of the linear algebraic equations which is derived from the homogeneous boundary value problem becomes very high as the total number of boundary knots increase which in turn affects the accuracy of the computation result combined with domain decomposition method we can avoid this problem easily in this paper we will solve homogeneous and non homogeneous partial differential equations using bkm combined with overlapped ddm from the numerical experiments we can find that it can solve pdes accurately

convergence analysis of krylov subspace iterations with methods from potential theory krylov subspace iterations are among the best known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices these methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix this leads to an extremal problem in polynomial approximation theory how small can a monic polynomial of a given degree be on the spectrum this survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices it is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure then depending on the number of iterations the lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region which is characterized by means of a constrained equilibrium problem from potential theory the same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems

a fast high resolution algorithm for linear convection problems particle transport method the paper is devoted to a novel explicit technique the particle transport method ptm for solving linear convection problems while being a lagrangian characteristic based method ptm has the advantage of eulerian methods to represent the solution on a fixed mesh the proposed approach belongs to the class of monotone high resolution numerical schemes possesses the property of unconditional stability and works with structured and unstructured meshes it is also demonstrated that the method has a linear computational complexity the performance of the presented algorithm is tested on one and two dimensional benchmark problems the numerical results confirm that the method has the 2nd order spatial accuracy and can be significantly faster than the grid based methods of the same order copyright 2006 john wiley sons ltd

rbf loi augmenting radial basis functions rbfs with least orthogonal interpolation loi for solving pdes on surfaces abstract we present a new method for the solution of pdes on manifolds m r d of co dimension one using stable scale free radial basis function rbf interpolation our method involves augmenting polyharmonic spline phs rbfs with polynomials to generate rbf finite difference rbf fd formulas these polynomial basis elements are obtained using the recently developed least orthogonal interpolation technique loi on each rbf fd stencil to obtain local restrictions of polynomials in r 3 to stencils on m the resulting rbf loi method uses cartesian coordinates does not require any intrinsic coordinate systems or projections of points onto tangent planes and our tests illustrate robustness to stagnation errors we show that our method produces high orders of convergence for pdes on the sphere and torus and present some applications to reaction diffusion pdes motivated by biology

coupling three field formulation and meshless mixed galerkin methods using radial basis functions in this work we solve the elliptic partial differential equation by coupling the meshless mixed galerkin approximation using radial basis function with the three field domain decomposition method the formulation has been adopted to increase the efficiency of the numerical technique by decreasing the error and dealing with the ill conditioning of the linear system caused by the radial basis function convergence analysis of the coupled technique is treated and numerical results of some solved examples are given at the end of this paper

localized radial basis functions based pseudo spectral method lrbf psm for nonlocal diffusion problems abstract spectral pseudo spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations in this paper we will present the use of a localized radial basis functions based pseudo spectral method lrbf psm for solving 2d nonlocal problems with radial nonlocal kernels the basic idea of the lrbf psm is to construct a set of orthogonal functions by rbfs on each overlapping sub domain from which the global solution can be obtained by extending the approximation on each sub domain to the entire domain numerical implementation indicates that the proposed lrbf psm is simple to use efficient and robust to solve various nonlocal problems

mesh free semi lagrangian methods for transport on a sphere using radial basis functions abstract we present three new semi lagrangian methods based on radial basis function rbf interpolation for numerically simulating transport on a sphere the methods are mesh free and are formulated entirely in cartesian coordinates thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface based coordinate systems for problems involving tracer transport in a given velocity field the semi lagrangian framework allows these new methods to avoid the use of any stabilization terms such as hyperviscosity during time integration thus reducing the number of parameters that have to be tuned the three new methods are based on interpolation using 1 global rbfs 2 local rbf stencils and 3 rbf partition of unity for the latter two of these methods we find that it is crucial to include some low degree spherical harmonics in the interpolants standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy efficiency conservation properties and dissipation dispersion errors for global rbfs spectral spatial convergence is observed for smooth solutions on quasi uniform nodes while high order accuracy is observed for the local rbf stencil and partition of unity approaches

a point interpolation meshless method based on radial basis functions a point interpolation meshless method is proposed based on combining radial and polynomial basis functions involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis this non singularity is useful in constructing well performed shape functions furthermore the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property this makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least squares approximation in addition the partial derivatives of shape functions are easily obtained thus improving computational efficiency examples on curve surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high copyright 2002 john wiley sons ltd

complete stagnation of gmres for normal matrices in this paper we study the problem of complete stagnation of the generalized minimum residual gmres method for normal matrices we first characterize all n i n nonsingular normal matrices a such that gmres a b stagnates completely for some vector b also we give necessary and sufficient conditions for the non existence of a real stagnation vector for real normal matrices the number of real stagnation vectors for normal matrices is studied moreover we characterize all the eigenvalues of nonsingular normal matrices a m 3 c such that gmres a b stagnates completely for some b c 3 using the results derived by a greenbaum v ptak and z strakos in 1996 we consider the complete stagnation of unitary matrices and derive another characterization for all nonsingular normal matrices a m 3 r such that gmres a b stagnates completely for some vector b r 3

convergence and error estimates for meshless galerkin methods in this paper the theory of radial basis functions together with galerkin methods is used to deal with the partial differential equations with dirichlet boundary conditions after general description the convergence and error estimates of this method are given in arbitrary dimensions some numerical examples are also given to illustrate the efficacious of the method

multicloud multigrid convergence with a meshless operator the primary objective of this work is to develop and test a new convergence acceleration technique we call multicloud multicloud is well founded in the mathematical basis of multigrid but relies on a meshless operator on coarse levels the meshless operator enables extremely simple and automatic coarsening procedures for arbitrary meshes using arbitrary fine level discretization schemes the performance of multicloud is compared with established multigrid techniques for structured and unstructured meshes for the euler equations on two dimensional test cases results indicate comparable convergence rates per unit work for multicloud and multigrid however because of its mesh and scheme transparency multicloud may be applied to a wide array of problems with no modification of fine level schemes as is often required with agglomeration techniques the implication is that multicloud can be implemented in a completely modular fashion allowing researchers to develop fine level algorithms independent of the convergence accelerator for complex three dimensional problems

radially symmetric weighted extended b spline model in this paper the weighted extended basis splines approach in the finite element method is applied to the electrostatic electromagnetic wave and bioheat problems for inhomogeneous boundary conditions and radially symmetric structures this new method which does not need mesh generation overcomes some of the drawbacks of using meshes and piecewise uniform or linear trial functions two dimensional radially symmetric electrostatic and electromagnetic wave equations are evaluated we also attempt to propose a three dimensional radially symmetric unexposed human eye model for simulating changes in corneal temperature using these new finite elements in conjunction with linear quadratic and cubic b splines our findings indicate that weighted extended basis spline solutions improve the standard finite element method the simulation results which are verified using the values reported in the literature point out to better efficiency in terms of the accuracy level

hyperviscosity based stabilization for radial basis function finite difference rbf fd discretizations of advection diffusion equations abstract we present a novel hyperviscosity formulation for stabilizing rbf fd discretizations of the advection diffusion equation the amount of hyperviscosity is determined quasi analytically for commonly used explicit implicit and implicit explicit imex time integrators by using a simple 1d semi discrete von neumann analysis the analysis is applied to an analytical model of spurious growth in rbf fd solutions that uses auxiliary differential operators mimicking the undesirable properties of rbf fd differentiation matrices the resulting hyperviscosity formulation is a generalization of existing ones in the literature but is free of any tuning parameters and can be computed efficiently to further improve robustness we introduce a simple new scaling law for polynomial augmented rbf fd that relates the degree of polyharmonic spline phs rbfs to the degree of the appended polynomial when used in a novel ghost node formulation in conjunction with the recently developed overlapped rbf fd method the resulting method is robust and free of stagnation errors we validate the high order convergence rates of our method on 2d and 3d test cases over a wide range of peclet numbers 1 1000 we then use our method to solve a 3d coupled problem motivated by models of platelet aggregation and coagulation again demonstrating high order convergence rates

solve partial differential equations by meshless subdomains method combined with rbfs abstract in this paper we will introduce a new meshless method for the numerical solutions of the partial differential equations it can be regarded as one of the petrov galerkin methods see zhang yun xin tan yong ji solve pdes using meshless petrov galerkin methods with rbfs submitted for publication in which the test space is generated by the radial basis functions rbfs and the trial space is generated by some characteristic functions but it is more efficient than the usual petrov galerkin methods because the complicated integrals in the usual petrov galerkin methods are replaced by the simple local boundary integrals the numerical examples indicate it is really an accurate and efficient meshless method certainly our method also can be regarded as one of the weighted residual methods or the generalization of the collocation methods

meshless method with ridge basis functions abstract meshless collocation methods for the numerical solutions of pdes are increasingly adopted due to their advantages including efficiency and flexibility and radial basis functions are popularly employed to represent the solutions of pdes motivated by the advantages of ridge basis function representation of a given function such as the connection to neural network fast convergence as the number of terms is increased better approximation effects and various applications in engineering problems a meshless method is developed based on the collocation method and ridge basis function interpolation this method is a truly meshless technique without mesh discretization it neither needs the computation of integrals nor requires a partition of the region and its boundary moreover the method is applied to elliptic equations to examine its appropriateness numerical results are compared to that obtained from other meshless methods and influence factors of accuracy for numerical solutions are analyzed

subdomain radial basis collocation method for fracture mechanics the direct approximation of strong form using radial basis functions rbfs commonly called the radial basis collocation method rbcm has been recognized as an effective means for solving boundary value problems nevertheless the non compactness of the rbfs precludes its application to problems with local features such as fracture problems among others this work attempts to apply rbcm to fracture mechanics by introducing a domain decomposition technique with proper interface conditions the proposed method allows 1 natural representation of discontinuity across the crack surfaces and 2 enrichment of crack tip solution in a local subdomain with the proper domain decomposition and interface conditions exponential convergence rate can be achieved while keeping the discrete system well conditioned the analytical prediction and numerical results demonstrate that an optimal dimension of the near tip subdomain exists the effectiveness of the proposed method is justified by the mathematical analysis and demonstrated by the numerical examples copyright 2010 john wiley sons ltd

stability and accuracy of three fourier expansion based strong form finite elements for the free vibration analysis of laminated composite plates summary in the present paper strong form finite elements are employed for the free vibration study of laminated arbitrarily shaped plates in particular the stability and accuracy of three different fourier expansion based differential quadrature techniques are shown these techniques are used to solve the partial differential system of equations inside each computational element the three approaches are called harmonic differential quadrature fourier differential quadrature and improved fourier expansion based differential quadrature methods the improved fourier expansion based differential quadrature method implements auxiliary functions in order to approximate functional derivatives up to the fourth order with respect to the fourier differential quadrature method that has a basis made of sines and cosines all the present applications are related to literature comparisons and the presentation of new results for further investigation within the same topic a study of such kind has never been proposed in the literature and it could be useful as a reference for future investigation in this matter copyright 2016 john wiley sons ltd

radial basis functions for the solution of hypersingular operators on open surfaces we analyze the approximation by radial basis functions of a hypersingular integral equation on an open surface in order to accommodate the homogeneous essential boundary condition along the surface boundary scaled radial basis functions on an extended surface and lagrangian multipliers on the extension are used we prove that our method converges quasi optimally approximation results for scaled radial basis functions indicate that for highly regular radial basis functions the achieved convergence rates are close to the one of low order conforming boundary element schemes numerical experiments confirm our conclusions

collocation discretizations of the transport equation with radial basis functions in this paper a new gridless method for numerically solving hyperbolic partial differential equations is presented this method uses collocation based on hermite interpolation at scattered sites at each time step the basis can be chosen at each time step which makes the approach adaptive and allows flexibility the goal of this paper is to demonstrate the potential of this adaptive flexible model which includes the ability to use scattered sites in a multi dimensional setting with compactly supported radial functions to illustrate the method and to compare it with known results we analyze a certain transport problem in detail when combined with an implicit time discretization and applied to smooth solutions we show that the method yields spectral rates of convergence in the spatial domain the implicit time discretization makes possible time steps of a size appropriate to the smoothness of the solution

solving partial differential equations by collocation using radial basis functions after a series of application papers have proven the approach to be numerically effective this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with possibly radial basis functions

grid free adaptive semi lagrangian advection using radial basis functions this paper proposes a new grid free adaptive advection scheme the resulting algorithm is a combination of the semi lagrangian method slm and the grid free radial basis function interpolation rbf the set of scattered interpolation nodes is subject to dynamic changes at run time based on a posteriori local error estimates a self adaptive local refinement and coarsening of the nodes serves to obtain enhanced accuracy at reasonable computational costs due to well known features of slm and rbf the method is guaranteed to be stable it has good approximation behaviour and it works for arbitrary space dimension numerical examples in two dimensions illustrate the performance of the method in comparison with existing grid based advection schemes

on the optimal shape parameters of radial basis functions used for 2 d meshless methods abstract a radial point interpolation meshless or radial pim method was proposed by authors to overcome the possible singularity associated with only polynomial basis the radial pim used multiquadric mq or gaussian as basis functions these two radial basis functions all included shape parameters although choice of shape parameters has been a hot topic in approximation theory and some empirical formulae were proposed how these shape parameters affect the accuracy of the radial pim has not been studied yet this paper studied the effect of shape parameters on the numerical accuracy of radial pim a range of suitable shape parameters is obtained from the analysis of the condition number of the system matrix error of energy and irregularity of node distribution it is observed that the widely used shape parameters for mq and reciprocal mq basis are not even close to their optimums the optimal shape parameters are found in this paper to be simply q 1 03 and r 1 42 for mq basis and c 0 003 0 03 for gaussian basis

max min properties of matrix factor norms given a set of real matrices c 0 c 1 ldots c k conditions are considered under which the equality mathop min limits alpha 1 ldots alpha k mathop max limits w 1 left left c 0 sum i 1 k alpha i c i right w right mathop max limits w 1 mathop min limits alpha 1 ldots alpha k left left c 0 sum i 1 k alpha i c i right w right holds it is shown that if the matrices c i i 0 1 ldots k are normal and commute with one another then the equality holds in particular this implies that if c i a i or c i a k i where a is a normal matrix then the equality holds an example is given to show that the equality may fail for noncommuting matrices when k 1 it is shown that the equality holds for arbitrary matrices if k 1

meshfree exponential integrators for the numerical solution of time dependent partial differential equations a class of meshfree exponential integrators is proposed these methods are of particular interest in situations where the solution of the differential equation concentrates on a small part of the computational domain which may vary in time for the space discretization radial basis functions with compact support are suggested the reasons for this choice are the stability and robustness of the resulting interpolation procedure the time integration is performed with an exponential rosenbrock method the required matrix functions are computed by newton interpolation based on leja points the proposed integrators are fully adaptive in space and time numerical examples that illustrate the robustness and the good stability properties of the method are included

a defect corrected finite element approach for the accurate evaluation of magnetic fields on unstructured grids in electromagnetic simulations of magnets and machines one is often interested in a highly accurate and local evaluation of the magnetic field uniformity based on local post processing of the solution a defect correction scheme is proposed as an easy to realize alternative to higher order finite element or hybrid approaches radial basis functions rbfs are key for the generality of the method which in particular can handle unstructured grids also contrary to conventional finite element basis functions higher derivatives of the solution can be evaluated as required e g for deflection magnets defect correction is applied to obtain a solution with improved accuracy and adjoint techniques are used to estimate the remaining error for a specific quantity of interest significantly improved local convergence orders are obtained the scheme is also applied to the simulation of a sterngerlach magnet currently in operation