boundary methods for solving elliptic problems with singularities and interfaces boundary approximation techniques are described for solving homogeneous self adjoint elliptic equations piecewise expansions into particular solutions are used which approximate both the boundary and interface conditions in a least squares sense convergence of such approximations is proved and error estimates are derived in a natural norm numerical experiments are reported for the singular motz problem which yield extremely accurate solutions with only a modest computational effort

element free galerkin method convergence of the continuous and discontinuous shape functions abstract we consider numerical solutions of second order elliptic partial differential equations such as laplace s equation or linear elasticity in two dimensional non convex domains by the element free galerkin method efg this is a meshless method in which the shape functions are constructed by using weight functions of compact support for non convex domains two approaches to the determination of whether a node affects approximation at a particular point are used a contained path criterion and the visibility criterion we show that for non convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions the resulting approximation is no longer conforming and its convergence must be established by inspection of the so called consistency term we show that the variant of the element free galerkin method which uses the discontinuous shape functions is convergent and that in the practically important case of linear shape functions the convergence rate is not affected by the discontinuities the convergence of the discontinuous approximation is first established by the classical and generalized patch test as these tests do not provide an estimate of the convergence rate the rate of convergence in the energy norm is examined for both the continuous and discontinuous efg shape functions and for smooth and non smooth solutions by a direct inspection of the error terms

a reproducing kernel method with nodal interpolation property a general formulation for developing reproducing kernel rk interpolation is presented this is based on the coupling of a primitive function and an enrichment function the primitive function introduces discrete kronecker delta properties while the enrichment function constitutes reproducing conditions a necessary condition for obtaining a rk interpolation function is an orthogonality condition between the vector of enrichment functions and the vector of shifted monomial functions at the discrete points a normalized kernel function with relative small support is employed as the primitive function this approach does not employ a finite element shape function and therefore the interpolation function can be arbitrarily smooth to maintain the convergence properties of the original rk approximation a mixed interpolation is introduced a rigorous error analysis is provided for the proposed method optimal order error estimates are shown for the meshfree interpolation in any sobolev norms optimal order convergence is maintained when the proposed method is employed to solve one dimensional boundary value problems numerical experiments are done demonstrating the theoretical error estimates the performance of the method is illustrated in several sample problems copyright 2003 john wiley sons ltd

a low cost goal oriented compact proper orthogonal decomposition basis for model reduction of static systems a novel model reduction technique for static systems is presented the method is developed using a goal oriented framework and it extends the concept of snapshots for proper orthogonal decomposition pod to include sensitivity derivatives of the state with respect to system input parameters the resulting reduced order model generates accurate approximations due to its goal oriented construction and the explicit training of the model for parameter changes the model is less computationally expensive to construct than typical pod approaches since efficient multiple right hand side solvers can be used to compute the sensitivity derivatives the effectiveness of the method is demonstrated on a parameterized aerospace structure problem

new boundary condition treatments in meshfree computation of contact problems due to the loss of kronecker delta properties in the meshfree shape functions the imposition of essential boundary conditions consumes significant cpu time in meshfree computation in this work two boundary condition treatments are proposed to enhance the computational efficiency of meshfree methods for contact problems the mixed transformation method is modified from the previous full transformation method by introducing a node partitioning and a mixed coordinate so that the matrix inversion and multiplication of coordinate transformation involves operations only on the sub degrees of freedom associated with the boundary group the boundary singular kernel method introduces singularities to the kernel functions of the essential and contact boundary nodes so that the corresponding coefficients of the singular kernel shape functions recover nodal values and consequently kinematic constraints can be imposed directly the effectiveness of the proposed methods is demonstrated in several numerical examples

reproducing kernel particle methods for large deformation analysis of non linear structures abstract large deformation analysis of non linear elastic and inelastic structures based on reproducing kernel particle methods rkpm is presented the method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties in large deformation analysis the current formulation considers hyperelastic and elasto plastic materials since they represent path independent and path dependent material behaviors respectively in this paper a material kernel function and an rkpm material shape function are introduced for large deformation analysis the support of the rkpm material shape function covers the same set of particles during material deformation and hence no tension instability is encountered in the large deformation computation the essential boundary conditions are introduced by the use of a transformation method the transformation matrix is formed only once at the initial stage if the rkpm material shape functions are employed the appropriate integration procedures for the moment matrix and its derivative are studied from the standpoint of reproducing conditions in transient problems with an explicit time integration method the lumped mass matrices are constructed at nodal coordinate so that masses are lumped at the particles several hyperelasticity and elasto plasticity problems are studied to demonstrate the effectiveness of the method the numerical results indicated that rkpm handles large material distortion more effectively than finite elements due to its smoother shape functions and consequently provides a higher solution accuracy under large deformation unlike the conventional finite element approach the nodal spacing irregularity in rkpm does not lead to irregular mesh shape that significantly deteriorates solution accuracy no volumetric locking is observed when applying non linear rkpm to nearly incompressible hyperelasticity and perfect plasticity problems further model adaptivity in rkpm can be accomplished simply by adding more points in the highly deformed areas without remeshing

numerical comparison of iterative eigensolvers for large sparse symmetric positive definite matrices abstract the jacobi davidson jd method has been recently proposed for the evaluation of the partial eigenspectrum of large sparse matrices in this work we report a set of numerical experiments that compare this method with other previously proposed techniques deflation accelerated conjugate gradient dacg and lanczos arpack on large sparse symmetric matrices the results obtained by jd and dacg are benchmarked against those obtained with arpack in terms of computational time for the evaluation of s s 100 leftmost eigenpairs the comparison is performed on a number of matrices with dimensions up to a few hundred thousands arising from the discretization of the diffusion equation by means of finite element mixed finite element and finite difference techniques the results show that dacg and jd are equally more efficient than arpack for a small number of eigenpairs with dacg more performing for small s and particularly for s 1 while arpack tends to become competitive when 40 or more eigenpairs are sought for the largest and worst conditioned matrix dacg proves the only viable alternative

wave patterns in an excitable reaction diffusion system abstract simulations of a non autonomous periodically forced excitable reaction diffusion system described by two parabolic partial differential equations with sh kinetics reveal the existence of various types of periodic and aperiodic wave trains the arrangement of the main entrainment regions in the forcing amplitude forcing period plane reflects the existence of both dynamic and static thresholds bursting phenomena are observed when the effects of the sinusoidal forcing at the boundary are studied

the method of parameter functionalization in the hopf bifurcation problem depending on a parameter for an extensive bibliography see l the greater part of these investigations exploit information about not only the linear terms of the right hand member of the system l the linear operator fl a 0 but also information about the terms of higher orders in the taylor expansion in x of the function f a x with the help of such information one can answer the questions on the number of arising self oscillations their stability their dependence on a parameter etc proofs of corresponding assertions use as a rule an analytic technique such as varied forms of the implicit function theorem the theory of invariant manifolds and the like at the same time in problems arising during the study of complicated physical technological etc processes often the only rather complete information available is that concerning the linear terms of the right hand member of the system 1 in such circumstances it is very difficult to apply an analytic technique for studying the problem of generation of self oscillations yet as things turn out 2 in some cases the very fact of generation of small self oscillations can be picked out of information about the linear terms of the right hand member of the system 1 of course under lack of information about the high order terms in the taylor expansion in x of f a x one can say next to nothing about properties of arising self oscillations this article contains a topological proof of the hopf theorem in this proof the method of parameter functionalization 3 introduced by krasnosel skii in another situation has much significance employment of topological considerations makes it possible to throw aside usual assumptions of differentiability of the right hand member of the system 1 this provides an opportunity to investigate systems l the right hand side of which contains for example hysteresis type or relay type nonlinearities the proof presented below goes through without changes in the case of functional differential equations with lagging arguments 4 we do not present here the exact formulations of such assertions because our purpose is the demonstration of the method in s 61 the method of parameter functionalization has been applied to the investigation of bifurcation of long periodic solutions of differential equations and mappings

on a one parameter family of delay population equations studied e g by seifert lo where b 0 and a is a positive parameter it is known that if b 1 then any positive solution x t of 1 1 converges to a b 1 as t a if b c 1 then a hopf bifurcation occurs at a critical value a of the parameter a searching the hopf bifurcation phenomenon it is equivalent to the investigation into the problem of showing the existence of nonconstant periodic solutions in particular a large part of this investigation has been devoted to problems arising from current models of population growth in certain biological contexts see for example cushing 3 41 simpson l 11 and the references therein one approach in this regard has been to demonstrate that under certain conditions nonconstant periodic solutions can bifurcate from a constant steady state in 1 1 this steady state is the point a b 1 if b 1 then a b 1 is the carrying capacity of the system the performance of hopf bifurcation can be guaranteed if the conditions imposed to the system are appropriate to lead to the sufficient conditions of the so called hopf bifurcation theorem namely the noncreasonance condition as well as the generic branching condition what these conditions are and how they can be applied to several situations the reader may consult for instance chafee 2 hale 5 chapter 111 hale et al 6 p 1071 kozjakin and krasnoselskii 7 and zeidler 12 part i section 8 15 and part iv section 79 91 the purpose of the present paper is to discuss the nature of the positive solutions of a delay differential equation of the general form

an efficient finite element method for treating singularities in laplace s equation we present a new finite element method for solving partial differential equations with singularities caused by abrupt changes in boundary conditions or sudden changes in boundary shape terms from the local solution supplement the ordinary basis functions in the finite element solution all singular contributions reduce to boundary integrals after a double application of the divergence theorem to the galerkin integrals and the essential boundary conditions are weakly enforced using lagrange multipliers the proposed method eliminates the need for high order integration improves the overall accuracy and yields very accurate estimates for the singular coefftcients it also accelerates the convergence with regular mesh refinement and converges rapidly with the number of singular functions although here we solve the laplace equation in two dimensions the method is applicable to a more general class of problems

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

multigrid methods for the computation of singular solutions and stress intensity factors i corner singularities we consider the poisson equation u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain with cracks multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed the convergence rate for the stress intensity factors is mathcal o h 3 2 in whenfel2 and mathcal o h 2 in whenfeh1 the convergence rate in the energy norm is mathcal o h 1 in in the first case and mathcal o h in the second case the costs of these multigrid methods are proportional to the number of elements in the triangulation the general case wherefehm is also discussed

the post processing approach in the finite element method part 1 calculation of displacements stresses and other higher derivatives of the displacements this is the first in a series of three papers in which we discuss a method for post processing a finite element solution to obtain high accuracy approximations for displacements stresses stress intensity factors etc rather than take the values of these quantities directly from the finite element solution we evaluate certain weighted averages of the solution over the entire region these yield approximations are of the same order of accuracy as the strain energy we obtain error estimates and also present some numerical examples to illustrate the practical effectiveness of the technique in the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

a corrected xfem approximation without problems in blending elements the extended finite element method xfem enables local enrichments of approximation spaces standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured in elements that blend the enriched areas with the rest of the domain problems arise in general these blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate a modification of the xfem approximation is proposed in this work the enrichment functions are modified such that they are zero in the standard elements unchanged in the elements with all their nodes being enriched and varying continuously in the blending elements all nodes in the blending elements are enriched the modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements the corrected xfem is applied to problems in linear elasticity and optimal convergence rates are achieved copyright 2007 john wiley sons ltd

solving laplacian problems with boundary singularities a comparison of a singular function boundary integral method with the p hp version of the finite element method we solve a laplacian problem over an l shaped domain using a singular function boundary integral method as well as the p hp finite element method in the former method the solution is approximated by the leading terms of the local asymptotic solution expansion and the unknown singular coefficients are calculated directly in the latter method these coefficients are computed by post processing the finite element solution the predictions of the two methods are discussed and compared with recent numerical results in the literature

forced excitations and excitable chaos in the photosensitive oregonator under periodic sinusoidal perturbations abstract the responses of an excitable system to external periodic perturbations are studied by using the photosensitive oregonator that is a three variable model of the photosensitive belousov zhabotinsky reaction the system is set at the vicinity of the subcritical hopf bifurcation point and sinusoidal perturbation is given by light 0 a 0 sin 2 the dynamic threshold for firing appears as a u shaped line in plane under high frequency perturbations the system shows a subthreshold period doubling cascade subthreshold chaotic responses excitable chaos and reaches firing chaos as a result of expansion of orbits around the fixed point

imposing essential boundary conditions in mesh free methods imposing essential boundary conditions is a key issue in mesh free methods the mesh free interpolation does not verify the kronecker delta property and therefore the imposition of prescribed values is not as straightforward as for the finite element method the aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in galerkin based mesh free methods special attention is paid to the mesh free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the galerkin weak form particular examples are used to analyze and compare their performance in different situations

optimal order nonnested multigrid methods for solving finite element equations iii on degenerate meshes in this paper we consider several model problems where finite element triangular meshes with arbitrarily small angles high aspect ratios are utilized to deal with anisotropy interfaces or singular perturbations the constant rate independent of the number of unknowns the smallest angle the interface discontinuity the singular perturbation parameter etc convergence of some special nonnested multigrid methods for solving the finite element systems on such degenerate meshes will be proved numerical data are provided to support the analysis in each case

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

the analysis of smoothers for multigrid algorithms the purpose of this paper is to provide a general technique for defining and analyzing smoothing operators for use in multigrid algorithms the smoothing operators considered are based on subspace decomposition and include point line and block versions of jacobi and gauss seidel iteration as well as generalizations we shall show that these smoothers will be effective in multigrid algorithms provided that the subspace decomposition satisfies two simple conditions in many applications these conditions are trivial to verify

interpolation method for adapting reduced order models and application to aeroelasticity reduced order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real time analysis although most reduced order models can operate in near real time their construction can be computationally expensive as it requires accumulating a large number of system responses to input excitations furthermore reduced order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation together these two issues underline the need for a fast and robust method for adapting precomputed reduced order models to new sets of physical or modeling parameters to this effect this paper presents an interpolation method based on the grassmann manifold and its tangent space at a point that is applicable to structural aerodynamic aeroelastic and many other reduced order models based on projection schemes this method is illustrated here with the adaptation of computational fluid dynamics based aeroelastic reduced order models of complete fighter configurations to new values of the freestream mach number good correlations with results obtained from direct reduced order model reconstruction high fidelity nonlinear and linear simulations are reported thereby highlighting the potential of the proposed reduced order model adaptation method for near real time aeroelastic predictions using precomputed reduced order model databases

coefficients of the singularities for elliptic boundary value problems on domains with conical points iii finite element methods on polygonal domains in the two first parts of this work rairo model math anal numer 24 1990 pp 27e52 rairo model math anal numer 24 1990 pp 343 367 formulas giving the coefficients arising in the singular expansion of the solutions of elliptic boundary value problems on nonsmooth domains are investigated now for the case of homogeneous strongly elliptic operators with constant coefficients on polygonal domains the solution of such problems by the finite element method is considered in order to approximate the solution or the coefficients different methods are used based on the expressions of the coefficients that were obtained in the first two parts the dual singular function method is also generalized

optimal order nonnested multigrid methods for solving finite element equations i on quasi uniform meshes we prove that the multigrid method works with optimal computational order even when the multiple meshes are not nested when a coarse mesh is not a submesh of the finer one the coarse level correction usually does not have the a projection property and does amplify the iterative error in some components nevertheless the low frequency components of the error can still be caught by the coarse level correction since the amplified highfrequency errors will be damped out by the fine level smoothing efficiently the optimal work order of the standard multigrid method can still be maintained however unlike the case of nested meshes a nonnested multigrid method with one smoothing does not converge in general no matter whether it is a v cycle or a w cycle method it is shown numerically that the convergence rates of nonnested multigrid methods are not necessarily worse than those of nested ones since nonnested multigrid methods accept quite arbitrarily related meshes we may then combine the efficiencies of adaptive refinements and of multigrid algorithms

model order reduction for meshfree solution of poisson singularity problems summary model order reduction mor techniques for enriched reproducing kernel meshfree methods are proposed for analysis of poisson problems with mild and strong singularities the employment of an integrated singular basis function method isbfm in conjunction with the selection of harmonic near tip asymptotic basis functions leads to a galerkin formulation in which the non smooth near tip basis functions appear only on the boundaries away from the singularity point this approach avoids the need of integrating the derivatives of non smooth functions near the singularity point and yields a discrete system that allows effective mor procedures under this framework a decomposed reduction method equipped with two distinct projections for smooth and non smooth parts of the finite dimensional space is proposed compared with the uniform reduction approach using a single projection operator the decomposed projection on isbfm discrete system preserves the singularity behavior of the fine scale solution in its lower dimensional approximation analytical error estimation and stability analysis show that isbfm with the decomposed projection can achieve better accuracy with only a slight increase of condition number compared with the uniform reduction approach in the reduced order solution of singularity problems the numerical tests validate the effectiveness of the proposed methods copyright 2014 john wiley sons ltd

bifurcation and chaos in a periodically stimulated cardiac oscillator abstract periodic stimulation of an aggregate of spontaneously beating cultured cardiac cells displays phase locking period doubling bifurcations and aperiodic chaotic dynamics at different values of the stimulation parameters this behavior is analyzed by considering an experimentally determined one dimensional poincare or first return map a simplified version of the experimentally determined poincare map is proposed and several features of the bifurcations of this map are described

the singular function boundary integral method for 3 d laplacian problems with a boundary straight edge singularity abstract three dimensional laplace problems with a boundary straight edge singularity caused by two intersecting flat planes are considered the solution in the neighbourhood of the straight edge can be expressed as an asymptotic expansion involving the eigenpairs of the analogous two dimensional problem in polar coordinates which have as coefficients the so called edge flux intensity functions efifs the efifs are functions of the axial coordinate the higher derivatives of which appear in an inner infinite series in the expansion the objective of this work is to extend the singular function boundary integral method sfbim developed for two dimensional elliptic problems with point boundary singularities g c georgiou l olson g smyrlis a singular function boundary integral method for the laplace equation commun numer methods eng 12 1996 127 134 for solving the above problem and directly extracting the efifs approximating the latter by either piecewise constant or linear polynomials eliminates the inner infinite series in the local expansion and allows the straightforward extension of the sfbim as in the case of two dimensional problems the solution is approximated by the leading terms of the local asymptotic solution expansion these terms are also used to weight the governing harmonic equation in the galerkin sense the resulting discretized equations are reduced to boundary integrals by means of the divergence theorem the dirichlet boundary conditions are then weakly enforced by means of lagrange multipliers the values of the latter are calculated together with the coefficients of the efifs the sfbim is applied to a test problem exhibiting fast convergence of order k 1 k being the order of the approximation of the efifs in the l 2 norm and leading to accurate estimates for the efifs

new estimates for multilevel algorithms including the v cycle the purpose of this paper is to provide new estimates for certain multilevel algorithms in particular we are concerned with the simple additive multilevel algorithm discussed recently together with j xu and the standard v cycle algorithm with one smoothing step per grid we shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity for example the theory applies to the standard multigrid vcycle on the l shaped domain or a domain with a crack and yields a uniform convergence rate we also prove uniform convergence rates for the multigrid v cycle for problems with nonuniformly refined meshes finally we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid v cycle algorithms

global bifurcation structure in periodically stimulated giant axons of squid abstract the responses of a squid giant axon fell into three categories phase locked quasi periodic and chaotic a bifurcation structure of the response patterns was obtained as a function parameters i it the current pulse intensity normalized by threshold current and t current pulse intervals the periodic responses appeared through either tangent bifurcation or type iii bifurcation parameters of i it and t the characteristics of the response patterns and the bifurcation structure could be qualitatively explained by considering the two effects of periodic current input on normal axons one is to act as a periodic external force and the other is to induce self oscillation in the axon both the response and bifurcation characteristics could be ascribed to a specific nonlinear interaction between the induced self oscillation and the periodic input

the post processing approach in the finite element method part 2 the calculation of stress intensity factors abstract in the context of a model problem the authors describe post processing techniques for the calculation of generalized stress intensity factors they discuss two broad classes of methods one involving an influence function and the other related to the well known energy release principle of fracture mechanics an error analysis is sketched and two numerical examples are given to illustrate the effectivity of the techniques this is the second in a series of three papers author

a singular function boundary integral method for laplacian problems with boundary singularities a singular function boundary integral method for laplacian problems with boundary singularities is analyzed in this method the solution is approximated by the truncated asymptotic expansion for the solution near the singular point and the dirichlet boundary conditions are weakly enforced by means of lagrange multiplier functions the resulting discrete problem is posed and solved on the boundary of the domain away from the point of singularity the main result of this paper is the proof of convergence of the method in particular we show that the method approximates the generalized stress intensity factors i e the coefficients in the asymptotic expansion at an exponential rate a numerical example illustrating the convergence of the method is also presented

the important modes of subsystems a moment matching approach substructure coupling methods such as the component mode synthesis cms method have been studied in structural dynamics analysis since 1960s the modes of subsystems associated with the lowest frequencies are typically retained in these methods in this paper we present a coupling matrix based mode selection scheme for the cms method referred to as the cms method this new scheme is derived using a moment matching principle defined on the interface between substructures it is compatible to the one in recently proposed optimal modal reduction omr method due to givoli et al the improvements of the cms method to the cms and omr methods are demonstrated by numerical examples from structural dynamics in both frequency and time domains copyright 2006 john wiley sons ltd

a numerical procedure for the determination of certain quantities related to the stress intensity factors in two dimensional elasticity abstract this work is devoted to the investigation of corner and interface singularities in linear elasticity and to the development of a numerical code that computes certain quantities associated with those singularities we compute several terms of the asymptotic expansion of the solution of the elasticity system near the corner or interface which is of the form knr nfn in particular the n and fn we treat our problem as a non linear eigenvalue problem and we use an approach based on the shooting method for boundary value problems to create an analytic function whose zeros are the n above the zeros of this function are then found using an iteration method and a global method to make sure that no zero was overlooked the program works for the general case of anisotropic non homogeneous wedge with general boundary conditions as well as in the case of crossing interfaces in an internal point we test the convergence accuracy robustness efficiency and reliability of the code from those tests it can be seen that the code works quite well for the majority of the applications

on the determination of higher order terms of singular elastic stress fields near corners in two dimensional elasticity stresses at reentrant corners exhibit singular behavior the stress field is of the form sigma sigma rm k iota r lambda 2 1 f i theta lambda i where r are polar coordinates centered at the tip of the corner andf i i are smooth functions for practical use of this series the eigenvalues i which are generally complex numbers are required in order of ascending real part the problem then is to find the roots of a transcendental equation eigenequation in the complex plane and arranged in order of ascending real part a theorem is proved on the number location and nature of the roots of this equation with the real part in fixed intervals of length excellent initial estimates of the real part of the complex roots become available and so are bounds within which single real roots exist this enables the determination of any number of roots in ascending order of real part the critical angles at which the eigenvalues change nature are also determined it is shown that for certain cases and for the symmetric mode of deformation the eigenvalue 1 does not represent a rigid body rotation therefore it has to be included in the series representation of the stresses the coefficientsk i can be determined by recently developed extraction techniques thus allowing complete determination of the elastic field and enabling its correlation with experimental data on brittle fracture crack initiation plastic zone estimation etc

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

on a delay differential equation for single specie population variations on considere l equation differentielle scalaire a retard n t n t a bn t n t 1 on suppose a b constantes positives et on etudie les solutions pour t 0 telles que n t n 0 t 1 t 0 avec n 0 t une fonction initiale donnee positive et continue on obtient des resultats sur la limitation et le comportement asymptotique des solutions

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

multigrid methods for variational problems further results in an earlier paper siam j numer anal 19 1982 pp 924 929 we developed a convergence theory for a class of multigrid methods applied to positive definite self adjoint differential boundary value problems where the multigrid process used w cycling and richardson s iteration in the present paper we extend this theory to include v cycling and more general relaxation schemes such as jacobi and arbitrarily ordered gauss seidel

comparison of two methods for the computation of singular solutions in elliptic problems we compare two numerical methods for the solution of elliptic problems with boundary singularities the first is the integrated singular basis function method isbfm a finite element method in which the solution is approximated by standard polynomial basis functions supplemented by the leading terms of the local singular solution expansion a double application of green s theorem reduces all galerkin integrals containing singular terms to boundary integrals with nonsingular integrands the originally essential boundary conditions are weakly enforced by means of lagrange multipliers the second method is a singular function boundary integral method which can be viewed as a modification of the isbfm the solution is approximated only by the leading terms of the local solution expansion the discretized equations are boundary integrals and the dimension of the problem is reduced by one the two methods are applied to the cracked beam problem giving very accurate estimates of the leading singular coefficients comparisons are made and their limitations are discussed

the singular function boundary integral method for an elastic plane stress wedge beam problem with a point boundary singularity the singular function boundary integral method sfbim is applied for the numerical solution of a 2 d laplace model problem of a perfectly elastic wedge beam under plane stress conditions the beam has a point boundary singularity it includes a curved boundary part and is subjected to non trivial distributed external loading the implemented solution method converges for this special model problem extremely fast the numerical estimates attained for the leading singular coefficients of the local asymptotic expansion and the stress and strain fields are highly accurate as verified by comparison with the available analytical solution

on the sharpness of certain local estimates for projections into finite element spaces influence of a re entrant corner in a plane polygonal domain with a reentrant corner consider a homogeneous dirichlet problem for poisson s equation au f with f smooth and the corresponding galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi uniform uniformly regular triangulations with the diameter of each element compara