the use of pde centres in the local rbf hermitian method for 3d convective diffusion problems in this work an extension is proposed to the local hermitian interpolation lhi method a meshless numerical method based on interpolation with small and heavily overlapping radial basis function rbf systems this extension to the lhi method uses interpolation functions which themselves satisfy the partial differential equation pde to be solved in this way a much improved reconstruction of partial derivatives can be obtained resulting in significantly improved accuracy in many cases the implementation algorithm is described and is validated via three convection diffusion reaction problems for steady and transient situations a crank nicolson implicit time stepping technique is used for the time dependent problems in the proposed approach a form of analytical upwinding is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function which includes the desired information about the convective velocity field the implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one dimensional travelling front problem at peclet numbers of 500 1000 2000 5000 and infinity which corresponds to a shock front in the case of infinity in addition the accuracy of the numerical scheme is validated against a one dimensional steady state solution exhibiting strong boundary layer effects and also against a steady and a transient three dimensional convection diffusion problem on irregular datasets all the test cases are validated against the corresponding analytical solutions finally the effect of various interpolation stencil configurations is investigated and some important limitations on local data centre distribution are identified

loop star decomposition of basis functions in the discretization of the efie a general and readily applicable scheme is presented for the determination of the basis functions that allow the decomposition of the surface current into a solenoidal part and a nonsolenoidal remainder the proposed approach brings into correspondence these two parts with two scalar functions and generates the known loop and star basis functions the completeness of the loop star basis is discussed employing the presented scheme the issue of the irrotational property of the nonsolenoidal functions is addressed

a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

on deriving a locally corrected nystrom scheme from a quadrature sampled moment method a novel high order method of moment procedure with quadrature point based discretization is presented the scheme is equivalent to a moment method employing smooth basis and testing functions applying a fixed point numerical quadrature approximation for the outer integral mapping the current to the quadrature points then leads to a formulation that is equivalent to that derived via the locally corrected nystrom method the convergence properties of the current density and rcs for smooth and singular geometries in two and three dimensions are also studied

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

on the equivalence of rwg method of moments and the locally corrected nystrom method for solving the electric field integral equation the first order locally corrected nystrom lcn method for the electric field integral equation is modified to ensure continuity of the current between triangular elements of the mesh rao wilton glisson rwg basis functions are used to create a conversion matrix from the lcn representation of the current to the rwg method of moments mom representation of the current in order to enforce continuity of current between adjacent triangular flat patches benefits of the method are two fold first it provides 4 reduction in degrees of freedom and removes unacceptable error levels in first order lcn implementations second the method can be viewed as a point based discretization of the rwg mom offering improved efficiency in its acceleration with the fast multipole algorithm

convergence of unsymmetric kernel based meshless collocation methods this paper proves convergence of variations of the unsymmetric kernel based collocation method introduced by kansa in 1986 since then this method has been very successfully used in many applications though it may theoretically fail in special situations and though it had no error bound or convergence proof up to now thus it is necessary to add assumptions or to make modifications our modifications prevent numerical failure by dropping strict collocation and allow a rigorous mathematical analysis proving error bounds and convergence rates these rates improve with the smoothness of the solution the domain and the kernel providing the trial spaces but they are currently not yet optimal and deserve refinement they are based on rates of approximation to the residuals by nonstationary meshless kernel based trial spaces and they are independent of the type of differential operator the results are applicable to large classes of linear problems in strong form provided that there is a smooth solution and the test and trial discretizations are chosen with some care our analysis does not require assumptions like ellipticity and it can be extended to ill posed problems

the approximation power of moving least squares a general method for near best approximations to functionals on r d using scattered data information is discussed the method is actually the moving least squares method presented by the backus gilbert approach it is shown that the method works very well for interpolation smoothing and derivatives approximations for the interpolation problem this approach gives mclain s method the method is near best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation the interpolation approximation in r d is shown to be a c function and an approximation order result is proven for quasi uniform sets of data points

numerical characterization of divergence conforming constrained basis functions for surface integral equations this paper presents an algebraic method of generating arbitrary order basis functions suitable for use in moment methods the basis functions are constructed by enforcing a set of suitable constraints on a more general set of functions leading to a matrix representation of the constraint condition the desired bases are then found through use of the singular value decomposition divergence conforming constrained bases are presented for quadrilateral meshes for use in electric and magnetic field integral equations appropriate constraints for enforcing normal continuity at cell boundaries are discussed a variety of problems are analyzed using the constrained bases and results show that the constrained bases exhibit exponential convergence for smooth structures and produce systems whose condition numbers grow relatively slowly with increasing basis order

stable computations with flat radial basis functions using vector valued rational approximations one commonly finds in applications of smooth radial basis functions rbfs that scaling the kernels so they are flat leads to smaller discretization errors however the direct numerical approach for computing with flat rbfs rbf direct is severely ill conditioned we present an algorithm for bypassing this ill conditioning that is based on a new method for rational approximation ra of vector valued analytic functions with the property that all components of the vector share the same singularities this new algorithm rbf ra is more accurate robust and easier to implement than the contour pade method which is similarly based on vector valued rational approximation in contrast to the stable rbf qr and rbf ga algorithms which are based on finding a better conditioned base in the same rbf space the new algorithm can be used with any type of smooth radial kernel and it is also applicable to a wider range of tasks including calculating hermite type implicit rbf fd stencils we present a series of numerical experiments demonstrating the effectiveness of this new method for computing rbf interpolants in the flat regime we also demonstrate the flexibility of the method by using it to compute implicit rbf fd formulas in the flat regime and then using these for solving poisson s equation in a 3 d spherical shell

unsymmetric meshless methods for operator equations a general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well posed linear operator equations is presented the results provide optimal convergence rates if the test and trial spaces satisfy a stability condition operators need not be elliptic and the problems can be posed in weak or strong form without changing the theory non stationary kernel based trial and test spaces are shown to fit into the framework disregarding the operator equation as a special case unsymmetric meshless kernel based methods solving weakly posed problems with distributional data are treated in some detail this provides a foundation of certain variations of the meshless local petrov galerkin technique of s n atluri and collaborators

effect of quintic nonlinearity on soliton collisions in optical fibers abstract we investigate the effect of quintic nonlinearity on collisions between solitons in different frequency channels in optical fibers we calculate the intensity and dynamics of the collision induced emitted radiation and changes in the soliton parameters due to the collision this is done by means of a perturbation theory with two small parameters the quintic nonlinearity coefficient q and the reciprocal of the inter channel frequency difference 1 we find that the amplitude of the collision induced emitted radiation is proportional to q this radiation can be described as originating from a variation in the kerr nonlinearity coefficient with respect to distance along the fiber we also find that the only other effect of quintic nonlinearity on the collision up to second order in the perturbation theory is an additional phase shift proportional to q the results obtained by our numerical simulations of the perturbed nonlinear schrodinger equation are in very good agreement with theory

kinks and solitons solutions for the generalized kdv equation with two power nonlinearities in this work the generalized kdv equation with two power nonlinearities is studied the tanh method and two sets of ansatze involving hyperbolic functions are introduced for analytic study of this equation new kinds of solitary wave solutions are formally derived

a meshless interpolation algorithm using a cell based searching procedure in this paper we propose a fast algorithm for bivariate interpolation of large scattered data sets it is based on the partition of unity method for constructing a global interpolant by blending radial basis functions as local approximants and using locally supported weight functions the partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cell based searching procedure more precisely we construct a cell structure which partitions the domain and strictly depends on the dimension of the subdomains thus providing a meaningful improvement in the searching process compared to the nearest neighbour searching techniques presented in allasia et al 2011 and cavoretto and de rossi 2010 2012 in fact this efficient algorithm and in particular the new searching procedure enable us a fast computation also in several applications where the amount of data to be interpolated is often very large up to many thousands or even millions of points analysis of computational complexity shows the high efficiency of the proposed interpolation algorithm this is also supported by numerical experiments

constrained locally corrected nystrom method a generalization of the locally corrected nystrom lcn discretization method is outlined wherein sparse transformations of the lcn system matrix are obtained via singular value decompositions of local constraint matrices the local constraint matrices are used to impose normal continuity of the currents across boundaries shared by mesh elements due to the method s simplicity and flexibility it is straightforward to develop high order constrained lcn clcn systems for different formulations and mesh element types numerical examples demonstrate the memory savings provided by the clcn method and its improved accuracy when applied to geometries with sharp edges it is also shown that the clcn method maintains the high order convergence of the lcn method and it eliminates the need to include line charges in nystrom based discretizations of formulations that involve the continuity equation

sparse matrix factorization using overlapped localizing logos modes on a shifted grid different types of local global solution logos modes have been proposed to develop fast direct solvers for electromagnetic wave problems while radiating logos modes are suitable for high frequency problems this paper focuses on the localizing logos modes which are suitable for low to mid frequency problems factorizations relying on non overlapped localizing logos nl logos modes are provably error controllable but fail to provide acceptable asymptotic computational efficiencies factorizations using overlapped localizing logos ol logos modes provide improved asymptotic computational performance but have been observed to suffer from significant overhead and in some cases poorer error control this paper describes a new procedure for obtaining the ol logos modes it is observed that the asymptotic efficiency is similar to previously reported ol logos schemes while providing stable error control for standard problem formulations the overhead associated with computing the ol logos modes is also reduced significantly

divergence conforming constrained basis functions for hexahedral volume elements this communication presents an algebraic method for constructing arbitrary order divergence conforming basis functions on hexahedral volume elements the appropriate constraints within and on the boundaries of elements are provided in particular the handling of faces where quantities are discontinuous is discussed the resulting bases are numerically characterized in terms of error convergence and system conditioning for a moment method discretization of the electric field volume integral equation for dielectric scatterers results show the accuracy of the proposed method as well as the low system matrix condition number that can be maintained as the basis order and mesh discretization are increased

balancing sparse matrices for computing eigenvalues abstract applying a permuted diagonal similarity transform dpap t d 1 to a matrix a before calculating its eigenvalues can improve the speed and accuracy with which the eigenvalues are computed this is often called balancing this paper describes several balancing algorithms for sparse matrices and compares them against each other and the traditional dense algorithm we first discuss our sparse implementation of the dense algorithm our code is faster than the dense algorithm when the density of the matrix is no more than approximately 5 and is much faster for large sparse matrices we next describe a set of randomized balancing algorithms for matrices that are not given explicitly i e given a vector x we can compute only ax and perhaps a t x we motivate these krylov based algorithms using perron frobenius theory results are given comparing the krylov based algorithms to each other and to the sparse and dense direct balancing algorithms looking at norm reduction running times and the accuracy of eigenvalues computed after a matrix is balanced we conclude that sparse balancing algorithms are efficient preconditioners for eigensolvers

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

maxwell s equations with arbitrary self action nonlinearity in a waveguiding theory guided modes and asymptotic of eigenvalues abstract the paper treats a nonlinear eigenvalue problem that describes propagation of a transverse magnetic wave in a plane dielectric waveguide having perfectly conducted walls at both sides the dielectric s permittivity is characterised by an arbitrary monotonically increasing self action nonlinearity the full set of guided modes is described by eigenvalues of the corresponding nonlinear maxwell operator with appropriate boundary conditions we give a comprehensive analysis of this problem and develop an original approach to study its solvability and properties of solutions several results about existence of the eigenvalues are proved their distribution and asymptotic are found zeros of the eigenfunctions and their location are also determined criterion of periodicity for the eigenfunctions is found comparison theorem is derived etc it is shown that unbounded nonlinearities lead to appearance of nonlinearised solutions this results in the existence of a novel guided regime that cannot be described within the framework of perturbation of linear guided wave regimes

high order divergence conforming constrained bases for triangular cells this paper presents an algebraic technique for generating arbitrary order divergence conforming bases for curvilinear triangular cells the bases are constructed by enforcing appropriate constraints on a linear combination of general functions and then extracting the desired bases using singular value decompositions koornwinder dubiner polynomials are chosen as the general function set basic constraints are presented to obtain divergence conforming bases and additional constraints are presented to further enforce the bases to be nedelec results from a variety of problems are given to show that the bases exhibit high order convergence and also produce systems that are relatively well conditioned compared to other basis sets

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

a two stage adaptive scheme based on rbf collocation for solving elliptic pdes abstract in this paper we present a new adaptive two stage algorithm for solving elliptic partial differential equations via a radial basis function collocation method our adaptive meshless scheme is based at first on the use of a leave one out cross validation technique and then on a residual subsampling method each of phases is characterized by different error indicators and refinement strategies the combination of these computational approaches allows us to detect the areas that need to be refined also including the chance to further add or remove adaptively any points the resulting algorithm turns out to be flexible and effective through a good interaction between error indicators and refinement procedures several numerical experiments support our study by illustrating the performance of our two stage scheme finally the latter is also compared with an efficient adaptive finite element method

on the infinitely many nonperturbative solutions in a transmission eigenvalue problem for maxwell s equations with cubic nonlinearity the paper focuses on a transmission eigenvalue problem for maxwell s equations with cubic nonlinearity that describes the propagation of transverse magnetic waves along the boundaries of a dielectric layer filled with nonlinear kerr medium using an original approach it is proved that even for small values of the nonlinearity coefficient the nonlinear problem has infinitely many nonperturbative solutions eigenvalues and eigenwaves whereas the corresponding linear problem always has a finite number of solutions this fact implies the theoretical existence of a novel type of eigenwaves that do not reduce to the linear ones in the limit in which the nonlinear coefficient reduces to zero asymptotic distribution of the eigenvalues is found periodicity of the eigenfunctions is proved the exact formula for the period is found and the zeros of the eigenfunctions are determined

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

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

on new triangle quadrature rules for the locally corrected nystrom method formulated on nurbs generated bezier surfaces in 3 d nonuniform rational b splines nurbs are the most widely used technique in today s geometric computer aided design systems for modeling surfaces combining the locally corrected nystrom lcn method with nurbs requires formulating lcn on both quadrilateral and triangular bezier surfaces as a typical nurbs generated bezier mesh includes elements of both the types while on quadrilateral elements the product of 1 d gaussian quadrature rules can be applied to lcn effectively gaussian integration rules available for triangles cannot efficiently be applied to lcn for two reasons first they do not possess the same number of quadrature points as the number of functions in a complete set of polynomial basis at an arbitrary order second they exacerbate the condition number of the resulting matrix equation at higher orders due to an increasing density of quadrature points near the edges and corners of triangles in this paper we construct a new set of quadrature rules for bezier triangles i e degenerate quadrilaterals based on the newton cotes equidistant quadrature rules and apply these rules to the lcn solution of the electric magnetic and combined field integral equations results show that the new family of quadrature rules overcomes both the aforementioned issues and can be applied to lcn effectively for orders from 0 to 9 inclusively

partition of unity extension of functions on complex domains abstract we introduce an efficient algorithm called partition of unity extension or pux to construct an extension of desired regularity of a function given on a complex multiply connected domain in 2d function extension plays a fundamental role in extending the applicability of boundary integral methods to inhomogeneous partial differential equations with embedded domain techniques overlapping partitions are placed along the boundaries and a local extension of the function is computed on each patch using smooth radial basis functions a trivially parallel process a partition of unity method blends the local extrapolations into a global one where weight functions impose compact support the regularity of the extended function can be controlled by the construction of the partition of unity function we evaluate the performance of the pux method in the context of solving the poisson equation on multiply connected domains using a boundary integral method and a spectral solver with a suitable choice of parameters the error converges as a tenth order method down to 10 14

study of mixed order basis functions for the locally corrected nystro spl uml m method a high order locally corrected nystro spl uml m lcn method employing the mixed order basis functions proposed by c 80al spl iota s 80kan and peterson is presented for the electromagnetic scattering by targets composed of both dielectric and conducting bodies an integral operator based on a combined field formulation for conducting surfaces and a mu spl uml ller formulation for dielectric surfaces is used it is found that for general scattering objects mixed order basis functions accelerate the convergence of the lcn solution can eliminate spurious charges and can significantly reduce the condition number of the impedance matrix

exact relationship between the locally corrected nystrom scheme and rwg moment method for the mixed potential integral equation the exact relationship between the rao wilton glisson rwg method of moments mom and the locally corrected nystrom lcn method for the mixed potential mp electric field integral equation efie is presented as an extension to our work where we established analogous exact relationship for solving the efie in its vector potential vp form it is shown that in order to achieve one such relationship for the mp efie the first and zeroth order lcn methods must be respectively used for the discretization of the vp and scalar potential terms of the mp efie the resulting numerical scheme is a point based rwg mom discretization of the mp efie via the nystrom method due to the mp formulation of the efie the proposed method establishes notably higher accuracy compared to either rwg mom or lcn discretizations of the efie in the vp form the increased accuracy is attributed to the analytical cancellation of the line charge contributions in the mp formulation as opposed to numerical cancellation inherent in the vp formulation of the efie the detailed study and explanations of the above cancellations is presented along with their impact on the accuracy of the respective schemes for both canonical and realistic scattering targets at different frequencies

computing eigenmodes ofelliptic operators using radial basis functions radial basis function rbf approximations have been successfully used to solve boundary value problems numerically we show that rbfs can also be used to compute eigenmodes of elliptic operators particular attention is given to the laplacian operator in two dimensions including techniques to avoid degradation of the solution near the boundaries for regions with corner singularities special functions must be added to the basis to maintain good convergence numerical results compare favorably to basic finite element methods

error analysis of the reproducing kernel particle method interest in meshfree or meshless methods has grown rapidly in recent years in solving boundary value problems bvps arising in mechanics especially in dealing with difficult problems involving large deformation moving discontinuities etc in this paper we provide a theoretical analysis of the reproducing kernel particle method rkpm which belongs to the family of meshfree methods one goal of the paper is to set up a framework for error estimates of rkpm we introduce the concept of a regular family of particle distributions and derive optimal order error estimates for rkp interpolants on a regular family of particle distributions the interpolation error estimates can be used to yield error estimates for rkp solutions of bvps

numerical simulation of reaction diffusion neural dynamics models and their synchronization desynchronization application to epileptic seizures abstract cognitive disorders especially epilepsy are closely linked with synchronization desynchronization of neurons in the brain in this paper the dynamical modeling and behavior analysis of a fitzhugh nagumo neuron and also synchronization control of a network of fitzhugh nagumo neurons which promise the understanding of cognitive processing are studied to numerically simulate these models and in order to overcome their difficulties such as computational complexity multi dimensionality non linearity having large spatial and temporal domains and also having a discrete initial data we propose the use of a strongly meshless technique based on wendland compactly supported radial basis function in conjunction with an operator splitting algorithm the main advantage of the proposed algorithm is its computational complexity for example o n x 1 3 n x 2 3 of the inversion of all matrices which is much lower than the complexity of pure radial basis functions method o n x 1 3 n x 2 3 of the inversion of matrices numerical experiments are presented showing that the proposed approaches are extremely accurate and fast

all well posed problems have uniformly stable and convergent discretizations this paper considers a large class of linear operator equations including linear boundary value problems for partial differential equations and treats them as linear recovery problems for functions from their data well posedness of the problem means that this recovery is continuous discretization recovers restricted trial functions from restricted test data and it is well posed or stable if this restricted recovery is continuous after defining a general framework for these notions this paper proves that all well posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well posedness of the problem and independent of the computational discretization provided that sufficiently many test data are used the solutions of discretized problems converge when enlarging the trial spaces and the convergence rate is determined by how well the data of the function solving the analytic problem can be approximated by the data of the trial functions this allows new and very simple proofs of convergence rates for generalized finite elements symmetric and unsymmetric kansa type collocation and other meshfree methods like meshless local petrov galerkin techniques it is also shown that for a fixed trial space weak formulations have a slightly better convergence rate than strong formulations but at the expense of numerical integration since convergence rates are reduced to those coming from approximation theory and since trial spaces are arbitrary this also covers various spectral and pseudospectral methods all of this is illustrated by examples

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

fast solution methods in electromagnetics various methods for efficiently solving electromagnetic problems are presented electromagnetic scattering problems can be roughly classified into surface and volume problems while fast methods are either differential or integral equation based the resultant systems of linear equations are either solved directly or iteratively a review of various differential equation solvers their complexities and memory requirements is given the issues of grid dispersion and hybridization with integral equation solvers are discussed several fast integral equation solvers for surface and volume scatterers are presented these solvers have reduced computational complexities and memory requirements

on the role of polynomials in rbf fd approximations ii numerical solution of elliptic pdes rbf generated finite differences rbf fd have in the last decade emerged as a very powerful and flexible numerical approach for solving a wide range of pdes we find in the present study that combining polyharmonic splines phs with multivariate polynomials offers an outstanding combination of simplicity accuracy and geometric flexibility when solving elliptic equations in irregular or regular regions in particular the drawbacks on accuracy and stability due to runge s phenomenon are overcome once the rbf stencils exceed a certain size due to an underlying minimization property test problems include the classical 2 d driven cavity and also a 3 d global electric circuit problem with the earth s irregular topography as its bottom boundary the results we find are fully consistent with previous results for data interpolation

stable evaluation of gaussian radial basis function interpolants we provide a new way to compute and evaluate gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter i e for flat kernels this work is motivated by the fundamental ideas proposed earlier by bengt fornberg and his coworkers however following mercer s theorem an l 2 mathbb r d rho orthonormal expansion of the gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by fornberg larsson and flyer and that is applicable in arbitrary space dimensions d in addition to obtaining an accurate approximation of the radial basis function interpolant using many terms in the series expansion of the kernel we also propose and investigate a highly accurate least squares approximation based on early truncation of the kernel expansion

a study of combined field formulations for material scattering for a locally corrected nystro spl uml m discretization a study of the error convergence and condition number of three integral equation formulations derived for penetrable material scattering objects the poggio miller chang harrington wu and tsai pmchwt the mu spl uml ller and the pmchwt formulations is presented for a variety of problems when discretized via a locally corrected nystro spl uml m method the pmchwt formulation is a first kind integral equation with a hypersingular operator the mu spl uml ller formulation leads to a second kind equation consisting of a diagonal term plus a compact operator this form is both frequency and mesh stable however unlike the pmchwt formulation the error grows with the refractive index the pmchwt formulation is in the form of a second kind equation but has a hypersingular term

higher order hierarchical legendre basis functions for electromagnetic modeling this paper presents a new hierarchical basis of arbitrary order for integral equations solved with the method of moments mom the basis is derived from orthogonal legendre polynomials which are modified to impose continuity of vector quantities between neighboring elements while maintaining most of their desirable features expressions are presented for wire surface and volume elements but emphasis is given to the surface elements in this case the new hierarchical basis leads to a near orthogonal expansion of the unknown surface current and implicitly an orthogonal expansion of the surface charge in addition all higher order terms in the expansion have two vanishing moments in contrast to existing formulations these properties allow the use of very high order basis functions without introducing ill conditioning of the resulting mom matrix numerical results confirm that the condition number of the mom matrix obtained with this new basis is much lower than existing higher order interpolatory and hierarchical basis functions as a consequence of the excellent condition numbers we demonstrate that even very high order mom systems e g tenth order can be solved efficiently with an iterative solver in relatively few iterations

soliton soliton interaction with dual power law nonlinearity abstract the intra channel collision of optical solitons with dual power law nonlinearity is studied in this paper by the aid of quasi particle theory the perturbation terms that are considered in this paper are of hamiltonian type the suppression of soliton soliton interaction in presence of these perturbation terms is acheived the numerical simulations support the quasi particle theory

a discrete least squares collocation method for two dimensional nonlinear time dependent partial differential equations abstract in this paper we develop regularized discrete least squares collocation and finite volume methods for solving two dimensional nonlinear time dependent partial differential equations on irregular domains the solution is approximated using tensor product cubic spline basis functions defined on a background rectangular interpolation mesh which leads to high spatial accuracy and straightforward implementation and establishes a solid base for extending the computational framework to three dimensional problems a semi implicit time stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem a key finding of our study is that the newly proposed mesh free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero both methods produce a large constrained least squares problem that must be solved at each time step in the advancement of the solution we have found that regularization yields a relatively well conditioned system that can be solved accurately using qr factorization an extensive numerical investigation is performed to illustrate the effectiveness of the present methods including the application of the new method to a coupled system of time fractional partial differential equations having different fractional indices in different irregularly shaped regions of the solution domain

modular fast direct electromagnetic analysis using local global solution modes local global solution logos modes provide a computationally efficient framework for developing fast direct solution methods for electromagnetic simulations logos modes provide a single basis for which both electromagnetic operators and their inverses are sparse with a controllable accuracy for fixed frequency modeling with h refinement efficient application of this observation is shown to provide efficient factorization algorithms for both integral equation and finite element methods in addition to providing fast direct solutions the logos framework is also shown to provide a new strategy for modular electromagnetic analysis the possibility of developing efficient direct solution strategies at high frequencies using the logos framework is also discussed