su n quantum yang mills theory in two dimensions a complete solution a closed expression of the euclidean wilson loop functionals is derived for pure yang mills continuum theories with gauge groups su n and u 1 and space time topologies r1 r1 and r1 s1 for the u 1 theory we also consider the s1 s1 topology the treatment is rigorous manifestly gauge invariant manifestly invariant under area preserving diffeomorphisms and handles all piecewise analytic loops in one stroke equivalence between the resulting euclidean theory and and the hamiltonian framework is then established finally an extension of the osterwalder schrader axioms for gauge theories is proposed these axioms are satisfied in the present model

a note on the efficient implementation of implicit methods for odes the use of implicit methods for odes e g implicit runge kutta schemes requires the solution of nonlinear systems of algebraic equations of dimension sm where m is the size of the continuous differential problem to be approximated usually the solution of this system represents the most time consuming section in the implementation of such methods consequently the efficient solution of this section would improve their performance in this paper we propose a new iterative procedure to solve such equations on sequential computers

convergence of finite difference methods for convection diffusion problems with singular solutions we are concerned with initial boundary value problems of convection diffusion equations in a square whose solutions have unbounded derivatives near the boundary by using finite difference approximations with respect to spatial variables and an implicit method with respect to the time variable it is shown that the numerical solution is convergent if the derivatives go to infinity under proper conditions furthermore the convergence of numerical solution can be accelerated if the mesh points are some functions of equidistant mesh points

classical solutions of nonlinear beam equations existence and stabilization in this paper we consider the large amplitude planar motion of a cantilevered beam with a feedback force and or torque proportional and acting opposite to the velocity angular velocity respectively applied at the free end we derive nonlinear equations of motion for such a beam based upon certain natural geometric assumptions we then prove the existence of classical solutions for these equations and show that all classical solutions are uniformly exponentially stable

projective techniques and functional integration for gauge theories a general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact hausdorff spaces the procedure is then applied to gauge theories to carry out integration over the non linear infinite dimensional spaces of connections modulo gauge transformations this method of evaluating functional integrals can be used either in the euclidean path integral approach or the lorentzian canonical approach a number of measures discussed are diffeomorphism invariant and therefore of interest to the connection dynamics version of quantum general relativity the account is pedagogical in particular prior knowledge of projective techniques is not assumed

constructing smoothing functions in smoothed particle hydrodynamics with applications this paper presents a general approach to construct analytical smoothing functions for the meshfree lagrangian and particle method of smoothed particle hydrodynamics the approach uses integral form of function representation and applies taylor series expansion to the sph function and derivative approximations the constructing conditions are derived systematically which not only interpret the consistency condition of the method but also describe the compact supportness requirement of the smoothing function examples of sph smoothing function are constructed including some existing ones with this approach a new quartic smoothing function with some advantages is constructed and is applied to the one dimensional shock problem and a one dimensional tnt detonation problem the good agreement between the sph results and those from other sources shows the effectiveness of the approach and the newly constructed smoothing function in numerical simulations

superconvergence of solution derivatives for the shortley weller difference approximation of poisson s equation part i smoothness problems the finite difference method fdm using the shortley weller approximation can be viewed as a special kind of the finite element methods fems using the piecewise bilinear and linear functions and involving some integration approximation when u c3 s i e u c3 0 s and f c2 s the superconvergence rate o h2 of solution derivatives in discrete h1 norms by the fdm is derived for rectangular difference grids where h is the maximal mesh length of difference grids used and the difference grids are not confined to be quasiuniform comparisons are made on the analysis by the maximum principle and the fem analysis conversions between the fdm and the linear and bilinear fems are discussed and numerical experiments are provided to support superconvergence analysis made

superconvergence of the shortley weller approximation for dirichlet problems abstract this paper presents a superconvergence property of the shortley weller s w approximation applied to the poisson type dirichlet problem in a bounded domain r 2 with the boundary this means that if the exact solution belongs to c 3 1 then the approximate solution obtained by the s w formula gives o h 3 accuracy at every grid point whose distance to is o h and o h 2 accuracy at other grid points where h denotes the equal mesh size in x and y directions the similar property holds for the case u c l 2 where l 0 or 1 and 0 1 stands for the holder exponent numerical examples are also given which illustrate our results

on solution uniqueness of elliptic boundary value problems in this paper we consider the problem of solution uniqueness for the second order elliptic boundary value problem by looking at its finite element or finite difference approximations we derive several equivalent conditions which are simpler and easier than the boundedness of the entries of the inverse matrix given in yamamoto et al t yamamoto s oishi q fang discretization principles for linear two point boundary value problems ii numer funct anal optim 29 2008 213 224 the numerical experiments are provided to support the analysis made strictly speaking the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem and this condition should be checked by solving the corresponding eigenvalue problems an application of the equivalent conditions is that we may discover the uniqueness simultaneously while seeking the approximate solutions of elliptic boundary equations

hadamard well posedness for a class of nonlinear shallow shell problems abstract this paper is concerned with the nonlinear shallow shell model introduced in 1966 by w t koiter in on the nonlinear theory of thin elastic shells iii nederl akad wetensch proc ser b 69 1966 33 54 section 11 and later studied in m bernadou j t oden an existence theorem for a class of nonlinear shallow shell problems j math pures appl 9 60 3 1981 285 308 we consider a version of this model which is based upon the intrinsic shell modeling techniques introduced by michel delfour and jean paul zolesio we show existence and uniqueness of both regular and weak solutions to the dynamical model and that the solutions are continuous with respect to the initial data while existence and uniqueness of regular solutions to nonlinear dynamic shell equations has been known full hadamard well posedness of weak solutions as shown in this paper is a new result which solves an old open problem in the field

local boundary element based a new finite difference representation for poisson equations abstract we present a new finite difference method for solving poisson s equation with the dirichlet boundary condition on a more general type of discretization for given domain based on the local boundary element method the method uses the piecewise linear approximation and produce a sparse linear system despite the use of boundary elements the discrete maximum principal is established without any angle condition for the discrete cells of the discretization the convergence behavior is comparable to that of standard finite difference methods on rectangle grids and equally super convergence property is attained on more general meshes when the solution u is in the function class c 2 c 3 0 1 also if u c 3 1 the standard o h 2 convergence is obtained numerical tests are given which illustrate our results

corrective meshless particle formulations for time domain maxwell s equations in this paper a meshless approximation of electromagnetic em field functions and relative differential operators based on particle formulation is proposed the idea is to obtain numerical solutions for em problems by passing up the mesh generation usually required to compute derivatives and by employing a set of particles arbitrarily placed in the problem domain the meshless smoothed particle hydrodynamics method has been reformulated for solving the time domain maxwell s curl equations the consistency of the discretized model is investigated and improvements in the approximation are obtained by modifying the numerical process corrective algorithms preserving meshless consistency are presented and successfully used test problems dealing with even and uneven particles distribution are simulated to validate the proposed methodology also by introducing a comparison with analytical solution

the poisson problem a comparison between two approaches based on sph method abstract in this paper two approaches to solve the poisson problem are presented and compared the computational schemes are based on smoothed particle hydrodynamics method which is able to perform an integral representation by means of a smoothing kernel function by involving domain particles in the discrete formulation the first approach is derived by means of the variational formulation of the poisson problem while the second one is a direct differential method numerical examples on different domain geometries are implemented to verify and compare the proposed approaches the computational efficiency of the developed methods is also studied

global existence and uniform decay for wave equation with dissipative term and boundary damping in this paper we prove the existence uniqueness and uniform stability of strong and weak solutions of the nonlinear wave equation u t email protected b x u t f u 0 in bounded domains with nonlinear damped boundary conditions given by email protected email protected g u t 0 with restrictions on function f u g u t and b x we prove the existence by means of the glerkin method and obtain the asymptotic behavior by using of the multiplier technique from the idea of kmornik and zuazua see 7

convergence of consistent and inconsistent finite difference schemes and an acceleration technique this paper states and generalizes in part some recent results on finite difference methods for dirichlet problems in a bounded domain which the author has obtained by himself or with coworkers after stating a superconvergence property of finite difference solution for the case where the exact solution u belongs to c4 it is remarked that such a property does not hold in general if u c4 next a convergence theorem is given for inconsistent schemes under some assumptions furthermore it is shown that the accuracy of the approximate solution can be improved by a coordinate transformation numerical examples are also given

superconvergent second order cartesian method for solving free boundary problem for invadopodia formation in this paper we present a superconvergent second order cartesian method to solve a free boundary problem with two harmonic phases coupled through the moving interface the model recently proposed by the authors and colleagues describes the formation of cell protrusions the moving interface is described by a level set function and is advected at the velocity given by the gradient of the inner phase the finite differences method proposed in this paper consists of a new stabilized ghost fluid method and second order discretizations for the laplace operator with the boundary conditions dirichlet neumann or robin conditions interestingly the method to solve the harmonic subproblems is superconvergent on two levels in the sense that the first and second order derivatives of the numerical solutions are obtained with the second order of accuracy similarly to the solution itself we exhibit numerical criteria on the data accuracy to get such properties and numerical simulations corroborate these criteria in addition to these properties we propose an appropriate extension of the velocity of the level set to avoid any loss of consistency and to obtain the second order of accuracy of the complete free boundary problem interestingly we highlight the transmission of the superconvergent properties for the static subproblems and their preservation by the dynamical scheme our method is also well suited for quasistatic hele shaw like or muskat like problems

superconvergence analysis of approximate boundary flux calculations certain projection post processing techniques have been proposed for computing the boundary flux for two dimensional problems e g see carey et al 5 in a series of numerical experiments on elliptic problems they observed that these post processing formulas for approximate fluxes were almost o h 2 accurate for linear triangular elements in this paper we prove that the computed boundary flux iso h 2 ln 1 h accurate in the maximum norm for the partial method of 5 if the solutionu h 3 then the boundary flux error iso h 3 2 in thel 2 norm

functional integration on spaces of connections suppose thatgis a compact connected lie group andp mis a smooth principalg bundle we define a cylinder function on the space a of smooth connections onpto be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves inm completing the algebra of cylinder functions in the sup norm we obtain a commutative c algebra fun a let a generalized measure on a be a bounded linear functional on fun a we construct a generalized measure 0on a that is invariant under all automorphisms of the bundlep not necessarily fixing the basem this result extends previous work which assumedmwas real analytic and used only piecewise analytic curves in the definition of cylinder functions as before any graph withnedges embedded inmdetermines a c subalgebra of fun a isomorphic toc gn and the generalized measure 0 fun a c restricts to the linear functional onc gn given by integration against normalized haar measure ongn our result implies that the group g of gauge transformations acts as unitary operators onl2 a the hilbert space completion of fun a in the norm f 2 0 f 2 1 2 using spin networks we construct explicit functions spanning the subspacel2 a g l2 a consisting of vectors invariant under the action of g

the shortley weller embedded finite difference method for the 3d poisson equation with mixed boundary conditions this paper describes a method for the solution of the 3d poisson equation subject to mixed boundary conditions on an irregularly shaped domain a finite difference method is used with the domain embedded in a rectangular grid quadratic treatment of the boundary conditions is shown to be necessary to obtain uniform error of o d 2 this contrasts with the dirichlet case where both quadratic and linear treatments give o d 2 error although the coefficient of error may be much larger for the linear case explicit error estimates demonstrating this behaviour are found for the 1d case with similar behaviour found in 2d and 3d numerical examples finally the extension of this approach to the n dimensional case is given where n 3

variational and momentum preservation aspects of smooth particle hydrodynamic formulations this paper presents a new variational framework for various existing smooth particle hydrodynamic sph techniques and presents a new corrected sph formulation the linear and angular momentum preserving properties of sph formulations are also discussed the paper will show that in general in order to preserve angular momentum the sph equations must correctly evaluate the gradient of a linear velocity field a corrected algorithm that combines kernel correction with gradient correction is presented the paper will illustrate the theory presented with several examples relating to simple free surface flows

solution of poisson s equation by analytical boundary element integration the solution of poisson s equation is essential for many branches of science and engineering such as fluid mechanics geosciences and electrostatics solution of two dimensional poisson s equations is carried out by bem based on galerkin vector method in which the integrals that appear in the boundary element method are expressed by analytical integration in this paper the galerkin vector method is developed for more general case than presented in the previous works the integrals are computed for constant and linear elements in bem by employing analytical integration in bem computation the numerical schemes and coordinate transformations can be avoided the presented method can also be used for the multiple domain case the results of the analytical integration are employed in bem code and the obtained analytical expression will be applied to several examples where the exact solution exists the produced results are in good agreement with the exact solution

global existence and uniform stabilization of a generalized dissipative klein gordon equation type with boundary damping in this paper we prove the existence uniqueness and uniform stability of strong and weak solutions of the nonlinear generalized klein gordon equation 1 1 1 see sec i in bounded domains with nonlinear damped boundary conditions given by 1 1 3 see sec i with some restrictions on function f u h u g ut and b x we prove the existence and uniqueness by means of nonlinear semigroup method and obtain the uniform stabilization by using the multiplier technique

local and global well posedness of semilinear reissner mindlin timoshenko plate equations abstract this is a study of local and global well posedness of nonlinearly perturbed reissner mindlin timoshenko plate equations this pde system represents an extension of the timoshenko beam model to plates and accounts for shear deformations the primary feature of the considered model is the interplay between nonlinear viscous interior damping and nonlinear source terms the main results verify local and global existence of solutions as well as their continuous dependence on the initial data in the appropriate function spaces moreover a blow up result is proved for solutions with negative initial energy

analysis of difference approximations to a singularly perturbed two point boundary value problem on an adaptively generated grid over the last few years there has been a significant growth in the use of adaptive grid methods for the numerical solution of differential equations with steep solutions little has been done however on the error analysis of adaptive methods in this paper we present an analysis for an upwind finite difference solution of a singular perturbation problem on a grid that is generated adaptively by equidistributing a monitor function based on the exact solution it is shown that the discrete solutions converge uniformly with respect to the perturbation parameter epsilon this epsilon uniform convergence is illustrated by numerical computations

on the support of the ashtekar lewandowski measure we show that the ashtekar isham extension overline a g of the configuration space of yang mills theories a g is topologically and measure theoretically the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices

a compact difference scheme for the biharmonic equation in planar irregular domains we present a finite difference scheme applicable to general irregular planar domains to approximate the biharmonic equation the irregular domain is embedded in a cartesian grid in order to approximate delta 2 phi at a grid point we interpolate the data on the irregular stencil by a polynomial of degree six the finite difference scheme is delta 2 q phi 0 0 where q phi is the interpolation polynomial the interpolation polynomial is not uniquely determined we present a method to construct such an interpolation polynomial and prove that our construction is second order accurate for a regular stencil m ben artzi j p croisille and d fishelov siam j sci comput 31 2008 pp 303 333 shows that the proposed interpolation polynomial is fourth order accurate we present some suitable numerical examples

numerical solution of a singularly perturbed two point boundary value problem using equidistribution analysis of convergence abstract adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near singular solutions adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers see for example mulholland et al siam j sci comput 19 4 1998 1261 1289 much is now being done in developing error analyes for methods that are based on adaptivity in this paper we present a rigorous error analysis for the solution of a singularly perturbed two point boundary value problem on a grid that is constructed adaptively from a knowledge of the exact solution the discrete solutions are generated by an upwind finite difference scheme and the grid is formed by equidistributing a monitor function based on arc length an error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter epsilon the epsilon uniform convergence is confirmed by numerical computations

boundary feedback stabilization of the undamped timoshenko beam with both ends free in this paper we study a boundary feedback system of a class of nonuniform undamped timoshenko beam with both ends free we give some sufficient conditions and some necessary conditions for the system to have exponential stability our method is based on the operator semigroup technique the multiplier technique and the contradiction argument of the frequency domain method

a new finite difference representation for poisson s equation on r3 from a contour integral abstract in this paper we propose a new finite difference representation for solving a dirichlet problem of poisson s equation on r 3 the key idea of the new approach is to represent the solution with a contour integral connecting the nodal values of each local domain centered at each isolated grid node which is based on the boundary integral equation on the local domain and calculate the contour integral using a piecewise linear interpolation of the solution a superconvergence of the scheme is analyzed using a maximum principle and a priori estimate for the finite difference operator the convergence behavior is comparable to that of standard finite difference methods on rectangle grids and a superconvergence property is attained when the solution u is in the function class c 2 c 3 0 u c 3 1 the standard o h 2 convergence is obtained

uniform stabilization of the dynamic elastica by boundary feedback it is proved that the dynamic elastica which is a nonlinear model describing the large deflecting motion of an inextensible beam can be uniformly exponentially stabilized by boundary velocity feedback

monotonicity of some perturbations of irreducibly diagonally dominant m matrices this paper presents a new result concerning the perturbation theory of m matrices we give the proof of a theorem showing that some perturbations of irreducibly diagonally dominant m matrices are monotone together with an explicit bound of the norm of the perturbation one of the assumptions concerning the perturbation matrix is that the sum of the entries of each of its row is nonnegative the resulting matrix is shown to be monotone although it may not be diagonally dominant and its off diagonal part may have some positive entries we give as an application the proof of the second order convergence of an non centered finite difference scheme applied to an elliptic boundary value problem

on the kirchhoff plates equations with thermal effects and memory boundary conditions in this paper we study the existence of weak and strong global solutions and uniform decay of the energy to the kirchhoff plates equations with thermal effect and memory conditions working at the boundary we show that the dissipation produced by the memory effect not depend on the present values of temperature gradient that is we show that the dissipation produced by memory effect is strong enough to produce exponential decay of the solution provided the relaxation functions also decays exponentially when the relaxation functions decays polynomially we show that the solution decays polynomially with the same rate

radial basis functions for solving differential equations high order numerical methods for solving differential equations are in general fairly sensitive to perturbations in their data a previously proposed radial basis function rbf method namely an integrated multiquadric scheme imq is applied to two point boundary value problems whose solutions exhibit thin boundary layers as frequently observed among rbf methods the matrices arising are ill conditioned in this paper to the point of numerical singularity the sensitivity of the method to perturbations and round off error is investigated and evidence is provided that perturbations are not nearly as strongly amplified as suggested by the large condition numbers of the matrices used in the computation

uniform stabilization of a nonlinear beam by nonlinear boundary feedback we consider the planar motion of a uniform prismatic beam of length l we want to derive a model that reflects the effect of stretching on bending which necessarily leads to nonlinear partial differential equations for the motion of the beam we will however assume that the constitutive equations for bending are linear this is in agreement with existing engineering literature see for example s and the bibliography therein it should be remarked that the effect of stretching on bending becomes significant if in particular a rigid rotation is superimposed on the motion we do not consider such a rotation here even though it could be handled within the present framework we assume that the beam in its reference state occupies the region described in rectangular coordinates by

long time dynamics of von karman semi flows with non linear boundary interior damping abstract we study von karman evolution equations with non linear dissipation and with partially clamped and partially free boundary conditions two distinctive mechanisms of dissipation are considered i internal dissipation generated by non linear operator and ii boundary dissipation generated by shear forces friction acting on a free part of the boundary the main emphasis is given to the effects of boundary dissipation under suitable hypotheses we prove existence of a compact global attractor and finiteness of its fractal dimension we also show that any solution is stabilized to an equilibrium and estimate the rate of the convergence which in turn depends on the behaviour at the origin of the functions describing the dissipation

stratification of the generalized gauge orbit space different versions for defining ashtekar s generalized connections are investigated depending on the chosen smoothness category for the paths and graphs the label set for the projective limit our definition covers the analytic case as well as the case of webs

a hybrid method and unified analysis of generalized finite differences and lagrange finite elements abstract finite differences finite elements and their generalizations are widely used for solving partial differential equations and their high order variants have respective advantages and disadvantages traditionally these methods are treated as different strong vs weak formulations and are analyzed using different techniques fourier analysis or green s functions vs functional analysis except for some special cases on regular grids recently the authors introduced a hybrid method called adaptive extended stencil fem or aes fem conley et al 2016 which combines features of generalized finite differences and lagrange finite elements to achieve second order accuracy over unstructured meshes however its analysis was incomplete due to the lack of existing mathematical theory that unifies the formulations and analysis of these different methods in this work we introduce the framework of generalized weighted residuals to unify the formulation of finite differences finite elements and aes fem in addition we propose a unified analysis of the well posedness convergence and mesh quality dependency of these different methods we also report numerical results with aes fem to verify our analysis we show that aes fem improves the accuracy of generalized finite differences while reducing the mesh quality dependency and simplifying the implementation of high order finite elements

the shortley weller scheme for variable coefficient two point boundary value problems and its application to tumor growth problem with heterogeneous microenvironment abstract the first half of this work develops and analyzes the shortley weller scheme or ghost fluid method with quadratic extrapolation for a two point boundary value problem with variable coefficients where the boundary points are not on the uniform mesh we prove that the local truncation error is first order convergent near the boundary but the solution is third order accurate near the boundary and second order accurate away from the boundary the second half of this work applies this numerical scheme to investigate the tumor growth problems in heterogeneous microenvironment we discover that the classic darcy s law tumor model can capture the chemotaxis property using variable nutrient diffusion rate and the haptotaxis mechanism through the variable extracellular matrix ecm permeability specifically the tumor tends to move to the regions with higher diffusion rate or lower ecm permeability

effective condition number for finite difference method for solving the linear algebraic equations ax b with the symmetric and positive definite matrix a from elliptic equations the traditional condition number in the 2 norm is defined by cond l n where l and n are the maximal and minimal eigenvalues of the matrix a respectively the condition number is used to provide the bounds of the relative errors from the perturbation of both a and b such a cond can only be reached by the worst situation of all rounding errors and all b for the given b the true relative errors may be smaller or even much smaller than the cond which is called the effective condition number in chan and foulser effectively well conditioned linear systems siam j sci statist comput 9 1988 963 969 and christiansen and hansen the effective condition number applied to error analysis of certain boundary collocation methods j comput appl math 54 1 1994 15 36 in this paper we propose the new computational formulas for effective condition number cond eff and define the new simplified effective condition number cond e for the latter we only need the eigenvector corresponding to the minimal eigenvalue of a which can be easily obtained by the inverse power method in this paper we also apply the effective condition number for the finite difference method for poisson s equation the difference grids are not supposed to be quasiuniform under a non orthogonality assumption the effective condition number is proven to be o 1 for the homogeneous boundary conditions such a result is extraordinary compared with the traditional cond o hmin 2 where hmin is the minimal meshspacing of the difference grids used for the non homogeneous neumann and dirichlet boundary conditions the effective condition number is proven to be o h 1 2 and o h 1 2hmin 1 respectively where h is the maximal meshspacing of the difference grids numerical experiments are carried out to verify the analysis made

the effective condition number applied to error analysis of certain boundary collocation methods abstract the boundary collocation method bcm is widely used in the engineering community for the numerical solution of linear boundary value problems it is commonly stated that the results computed by this method are affected by large rounding errors this conclusion is based on a simple worst case analysis using the conventional condition number for linear systems in this paper we present a refined error analysis for a class of boundary value problems using the concept of effectively well conditioned systems this analysis leads to much smaller a priori bounds for the rounding error in the computed bcm solution these smaller bounds imply that the numerically computed solution expresses much more accurately than previously supposed properties of the ideal bcm solution computed with infinite precision therefore the intrinsic properties of the bcm can be investigated through numerical computations to a far larger extent than previously supposed thus allowing for a numerical diagnostic investigation of the bcm for example when the bcm is applied to problems with certain geometries it is observed that the approximation does not converge using our new smaller error bounds we conclude that the lack of convergence of the bcm solution for these geometries is due to the bcm itself and not to rounding errors

a stable sph discretization of the elliptic operator with heterogeneous coefficients smoothed particle hydrodynamics sph has been extensively used to model high and low reynolds number flows free surface flows and collapse of dams study pore scale flow and dispersion elasticity and thermal problems in different applications it is required to have a stable and accurate discretization of the elliptic operator with homogeneous and heterogeneous coefficients in this paper the stability and approximation analysis of different sph discretization schemes traditional and new of the diagonal elliptic operator for homogeneous and heterogeneous media are presented the optimum and new discretization scheme of specific shape satisfying thetwo point flux approximation nature is also proposed this scheme enhances the laplace approximation brookshaw s scheme 1985 and schwaiger s scheme 2008 used in the sph community for thermal viscous and pressure projection problems with an isotropic elliptic operator the numerical results are illustrated by numerical examples where the comparison between different versions of the meshless discretization methods are presented

a corrective smoothed particle method for boundary value problems in heat conduction combining the kernel estimate with the taylor series expansion is proposed to develop a corrective smoothed particle method cspm this algorithm resolves the general problem of particle deficiency at boundaries which is a shortcoming in standard smoothed particle hydrodynamics ssph in addition the method s ability to model derivatives of any order could make it applicable for any time dependent boundary value problems an example of the applications studied in this paper is unsteady heat conduction which is governed by second order derivatives numerical results demonstrate that besides the capability of directly imposing boundary conditions the present method enhances the solution accuracy not only near or on the boundary but also inside the domain published in 1999 by john wiley sons ltd this article is a u s government work and is in the public domain in the united states

the trefftz method using fundamental solutions for biharmonic equations in this paper the trefftz method of fundamental solution fs called the method of fundamental solution mfs is used for biharmonic equations the bounds of errors are derived for the mfs with almansi s fundamental solutions denoted as the mafs in bounded simply connected domains the exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions respectively the stability analysis of the mafs is also made for circular domains numerical experiments are carried out for both smooth and singularity problems the numerical results coincide with the theoretical analysis made when the particular solutions satisfying the biharmonic equation can be found the method of particular solutions mps is always superior to the mfs and the mafs based on numerical examples however if such singular particular solutions near the singular points do not exist the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points based on the computed results the mfs using the greedy adaptive techniques may provide more accurate solutions for singularity problems moreover the numerical solutions by the mafs with almansi s fs are slightly better in accuracy and stability than those by the traditional mfs hence the mafs with the afs is recommended for biharmonic equations due to its simplicity

the conditioning of some numerical methods for first kind boundary integral equations we consider the sensitivity of the matrix equations which arise when solving boundary integral equations of negative order specifically in case of the typical example of the operator with logarithmic kernel the sensitivity is expressed using condition numbers and it turns out that the local condition number is a very good indicator in contrast to the ordinary condition number traditionally used

analyses on the finite difference method by gibou et al for poisson equation gibou et al in 4 introduced a finite difference method for solving the poisson equation in irregular domains with the dirichlet boundary condition contrary to its great importance its properties have not been mathematically analyzed but have just been numerically observed in this article we present two analyses for the method one proves that its solution is second order accurate and the other estimates the condition number of its linear system according to our estimation the condition number of the unpreconditioned linear system is of size o 1 h h min and each of jacobi sgs and ilu preconditioned systems is of size o h 2 furthermore our analysis shows that the condition number of milu is of size o h 1 the most successful one

uniform stabilizability of a full von karman system with nonlinear boundary feedback full von karman system accounting for in plane accelerations and describing the transient deformations of a thin elastic plate subject to edge loading is considered the energy dissipation is introduced via the nonlinear velocity feedback acting on a part of the edge of the plate it is known j puel and m tucsnak siam j control optim 33 1995 pp 255 273 that in the case of linear dissipation and star shaped domains boundary velocity feedback with the tangential derivatives of horizontal displacements leads to the exponential decay rates for the energy of the resulting closed loop system the main goal of the paper is to derive the uniform energy decay rates valid for the model without the above mentioned restrictions in particular it is shown that simple monotone nonlinear feedback without the tangential derivatives of the horizontal displacements provides the uniform decay rates for the energy in the absence of geometric hypotheses imposed on the controlled part of the boundary this is accomplished by establishing among other things sharp regularity results valid for the boundary traces of solutions corresponding to this nonlinear model and by employing a holmgren type uniqueness result proved recently in v isakov j differential equations 97 1997 pp 134 147 for the dynamical systems of elasticity which are overdetermined on the boundary

penalty combinations of the ritz galerkin and finite difference methods for singularity problems abstract penalty combination of the ritz galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities the superconvergence rate o h2 of solution derivatives by the combination can be achieved while using quasiuniform rectangular difference grids where h is the maximal mesh length of difference grids used in the finite difference method and 0 is an arbitrarily small number it is due to its simplicity that the penalty combination of the ritz galerkin and finite difference methods is highly recommended for solving the complicated problems with multiple singularities and multiple interfaces

a simple adaptive grid method in two dimensions this paper gives an interpretation of the concept of equidistribution in the context of adaptive grid generation for multidimensional problems it is shown that the equidistribution principle cannot be satisfied throughout the domain of the problem and based on this recognition a local equidistribution principle is developed a discrete formulation is described for grid generation in two space dimensions and a smoothing mechanism is presented for improving mesh quality the adaptive grid method that is constructed contains three grid quality parameters numerical examples illustrate adaptive grid generation using a prescribed monitor function and grid generation for numerical solution of partial differential equations results show that the method produces high quality grids and that it is fairly insensitive to the choice of parameters

limit behavior of the solution to nonlinear viscoelastic marguerre von karman shallow shell system abstract this paper is concerned with the nonlinear full marguerre von karman shallow shell system with a dissipative mechanism of memory type the model depends on one small parameter the main purpose of this paper is to show that as the parameter approaches zero the limiting system is the well known full von karman model with memory for thin plates

an analysis of 1 d smoothed particle hydrodynamics kernels in this paper the smoothed particle hydrodynamics sph kernel is analyzed resulting in measures of merit for one dimensional sph various methods of obtaining an objective measure of the quality and accuracy of the sph kernel are addressed since the kernel is the key element in the sph methodology this should be of primary concern to any user of sph the results of this work are two measures of merit one for smooth data and one near shocks the measure of merit for smooth data is shown to be quite accurate and a useful delineator of better and poorer kernels the measure of merit for non smooth data is not quite as accurate but results indicate the kernel is much less important for these types of problems in addition to the theory 20 kernels are analyzed using the measure of merit demonstrating the general usefulness of the measure of merit and the individual kernels in general it was decided that bell shaped kernels perform better than other shapes

chebyshev tau matrix method for poisson type equations in irregular domain in this paper a new meshless method chebyshev tau matrix method ctmm is researched the matrix representations for the differentiation and multiplication of chebyshev expansions make ctmm easy to implement problems with curve boundary can be efficiently treated by ctmm poisson type problems including standard poisson problems helmholtz problems problems with variable coefficients and nonlinear problems are computed some numerical experiments are implemented to verify the efficiency of ctmm and numerical results are in good agreement with the analytical one it appears that ctmm is very effective for poisson type problems in irregular domains

stratification of the orbit space in gauge theories the role of nongeneric strata gauge theory is a theory with constraints and for that reason the space of physical states is not a manifold but a stratified space orbifold with singularities the classification of strata for smooth and generalized connections as well as the formulation of the physical space as the zero set of a momentum map is reviewed several important features of nongeneric strata are discussed and new results are presented suggesting an important role for these strata as concentrators of the measure in ground state functionals and as a source of multiple structures in low lying excitations

the immersed interface technique for parabolic problems with mixed boundary conditions a finite difference scheme is presented for a parabolic problem with mixed boundary conditions we use an immersed interface technique to discretize the neumann condition and we use the shortley weller approximation for the dirichlet condition the proof of a discrete maximum principle is given as well as the proof of convergence of the scheme this convergence is also validated on numerical examples

perturbation and stability analysis of strong form collocation with reproducing kernel approximation solving partial differential equations using strong form collocation with nonlocal approximation functions such as orthogonal polynomials and radial basis functions offers an exponential convergence but with the cost of a dense and ill conditioned linear system in this work the local approximation functions based on reproducing kernel approximation are introduced for strong form collocation method called the reproducing kernel collocation method rkcm we perform the perturbation and stability analysis of rkcm and estimate the condition numbers of the discrete equation our stability analyses validated with numerical tests show that this approach yields a well conditioned and stable linear system similar to that in the finite element method we also introduce an effective condition number where the properties of both matrix and right hand side vector of a linear system are taken into consideration in the measure of conditioning we first derive the effective condition number of the linear systems resulting from rkcm and show that using the effective condition number offers a tighter estimation of stability of a linear system the mathematical analysis also suggests that the effective condition number of rkpm does not grow with model refinement the numerical results are also presented to validate the mathematical analysis copyright 2011 john wiley sons ltd

a node centered local refinement algorithm for poisson s equation in complex geometries this paper presents a method for solving poisson s equation with dirichlet boundary conditions on an irregular bounded three dimensional region the method uses a nodal point discretization and adaptive mesh refinement amr on cartesian grids and the amr multigrid solver of almgren the discrete laplacian operator at internal boundaries comes from either linear or quadratic shortley weller extrapolation and the two methods are compared it is shown that either way solution error is second order in the mesh spacing error in the gradient of the solution is first order with linear extrapolation but second order with shortley weller examples are given with comparison with the exact solution the method is also applied to a heavy ion fusion accelerator problem showing the advantage of adaptivity

uniform stability in structural acoustic models with flexible curved walls abstract the aim of this paper is twofold first we develop an explicit extension of the kirchhoff model for thin shells based on the model developed by michel delfour and jean paul zolesio this model relies heavily on the oriented distance function which describes the geometry once this model is established we investigate the uniform stability of a structural acoustic model with structural damping the result no longer requires that the active wall be a plate it can be virtually any shell provided that the shell is thin enough to accommodate the curvatures

an infinite dimensional calculus for generalized connections on hypercubic lattices a space for gauge theories is defined using projective limits as subsets of cartesian products of homomorphisms from a lattice on the structure group in this space non interacting and interacting measures are defined as well as functions and operators from projective limits of test functions and distributions on products of compact groups a projective gauge triplet is obtained which provides a framework for the infinite dimensional calculus in gauge theories the gauge measure behavior on non generic strata is also obtained

uniqueness theorem for weak solutions of von karman evolution equations we prove a uniqueness theorem for weak solutions of the oscillation problem for von karman plates

on a global superconvergence of the gradient of linear triangular elements abstract we study a simple superconvergent scheme which recovers the gradient when solving a second order elliptic problem in the plane by the usual linear elements the recovered gradient globally approximates the true gradient even by one order of accuracy higher in the l 2 norm than the piecewise constant gradient of the ritz galerkin solution a superconvergent approximation to the boundary flux is presented as well

the embedded finite difference method for the poisson equation in a domain with an irregular boundary and dirichlet boundary conditions the poisson equation subject to dirichlet boundary conditions on an irregular domain can be treated by embedding the region in a rectangular domain and solving using finite differences over the rectangle the crucial issue is the discretization of the boundaries of the irregular domain in the past both linear and quadratic boundary treatments have been used and error bounds have been derived in both cases showing that the linear case gives uniform second order accuracy whereas the quadratic case gives third order accuracy at the boundaries and second order accuracy internally thus it has been recommended that the linear boundary treatment be used as it is simpler gives rise to a symmetric matrix formulation and has uniform accuracy the present work shows that this argument is inadequate because the coefficients of the error terms also play an important role we demonstrate this in the 1 d case by determining explicit expressions for the error for both the linear and quadratic boundary treatments it is shown that for the linear case the coefficient of error is in general large enough to dominate the calculation and that therefore it is necessary to use a quadratic boundary treatment in order to obtain errors comparable with those obtained for a regular domain we go on to show that the 1 d expressions for error can be used to approximate the boundary error for 2 d problems and that for the linear treatment the boundary error again dominates

error analysis of the method of fundamental solutions for linear elastostatics for linear elastostatics in 2d the trefftz methods i e the boundary methods using the particular solutions and the fundamental solutions satisfying the cauchy navier equation lead to the method of particular solutions mps and the method of fundamental solutions mfs respectively in this paper the mixed types of the displacement and the traction boundary conditions are dealt with and both the direct collocation techniques and the lagrange multiplier are used to couple the boundary conditions the former is just the mfs and the mps and the latter is also called the hybrid trefftz method htm in jirousek 1978 1992 1996 1 3 in bogomolny 1985 4 and li 2009 5 the error analysis of the mfs is given for laplace s equation and in li 2012 6 the error bounds of both mps and htm using particular solutions ps are provided for linear elastostatics in this paper our efforts are devoted to explore the error analysis of the mfs and the htm using fundamental solutions fs the key analysis is to derive the errors between fs and ps of the linear elastostatics where the expansions of the fs in li et al 2011 7 are a basic tool in analysis then the optimal convergence rates can be achieved for the mfs and the htm using fs recently the mfs has been developed with numerous reports in computation the analysis is behind the analysis of the mfs for linear elastostatics in this paper may narrow the existing gap between computation and theory of the mfs

the existence of smooth solutions for a nonlinear beam with boundary input and output in this article a dynamical system modeling the bending vibrations of a quasilinear beam is considered where the nonlinearity comes from hooke s law we are concerned with the existence of local smooth solutions to this quasi linear beam with boundary input and output applying the evolution system theory we first show that a related linear system admits a unique smooth solution then some energy estimates for the linear system are established finally the existence of local smooth solutions to the quasi linear system is shown through fixed point arguments

uniform decay rates for nonlinear viscoelastic marguerre von karman equations abstract in this paper we consider a model of nonlinear viscoelastic shallow shell that is referred to as the full marguerre von karman under the presence of long time memory we show that the energy functional associated with the system decays exponentially to zero as time goes to infinity

a study of monitor functions for two dimensional adaptive mesh generation in this paper we study the problem of two dimensional adaptive mesh generation using a variational approach and specifically the effect that the monitor function has on the resulting mesh behavior the basic theoretical tools employed are green s function for elliptic problems and the eigendecomposition of symmetric positive definite matrices based upon this study a general strategy is suggested for how to choose the monitor function and numerical results are presented for illustrative purposes the three dimensional case is also briefly discussed it is noted that the strategy used here can be applied to other elliptic mesh generation techniques as well

a simplified approach to enhance the performance of smooth particle hydrodynamics methods this paper presents an approach to improve the accuracy and stability of smooth particle hydrodynamics sph techniques and describes how this correction procedure can be combined with kernel correction to formulate a complete form of the corrected smooth particle hydrodynamics csph method the development and theory of this new approach is presented in the context of the solution of poisson equations successful implementation of the theory is verified via various numerical examples

superconvergence phenomenon in the finite element method arising from averaging gradients we study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements the averaged gradient is a piecewise linear continuous vector field the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point the convergence rate of the averaged gradient to an exact gradient in thel 2 norm can locally be higher even by one than that of the original piecewise constant discrete gradient