multiple nodal bound states for a quasilinear schrodinger equation nehari techniques are used to prove the existence of multiple indeed infinitely many nodal type bound states for the quasilinear schrodinger equation izt v0z z q x z 2z k z 2 z 0 with prescribed number of nodes

convergence of nonconforming multigrid methods without full elliptic regularity we consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity we prove that there is a bound 1 for the contraction number of the w cycle algorithm which is independent of mesh level provided that the number of smoothing steps is sufficiently large we also show that the symmetric variable v cycle algorithm is an optimal preconditioner

multilevel preconditioning for partition of unity methods some analytic concepts this paper is concerned with the construction and analysis of multilevel schwarz preconditioners for partition of unity methods applied to elliptic problems we show under which conditions on a given multilevel partition of unity hierarchy mpum one even obtains uniformly bounded condition numbers and how to realize such requirements the main anlytical tools are certain norm equivalences based on two level splits providing frames that are stable under taking subsets

a note on convergence of the multigrid v cycle several recent papers have discussed the convergence of the multigrid v cycle in particular there are several results for the symmetric case where the numbers of smoothings before the fine to coarse transfer and after the coarse to fine transfer are the same in most instances the smoother h i e 1a has been limited to the case where e is positive definite and the eigenvalues h of h satisfy 0

a p cycle analysis of multigrid method we suggest a new strategy for multigrid iterations based on our analysis for p cycle multigrid method first we improve a recent result on multigrid v cycle analysis given by bramble and pasciak to p cycle multigrid method we show that the convergence rate improves slowly with p eventually bounded by a constant which is in agreement with many previous experiments based on the this observation we use p cycle only in the first few multigrid iterations which is more effective than the standard p cycle

soliton solutions for quasilinear schrodinger equations the critical exponential case abstract quasilinear elliptic equations in r 2 of second order with critical exponential growth are considered by using a change of variable the quasilinear equations are reduced to semilinear equations whose respective associated functionals are well defined in h 1 r 2 and satisfy the geometric hypotheses of the mountain pass theorem using this fact we obtain a cerami sequence converging weakly to a solution v in the proof that v is nontrivial the main tool is the concentration compactness principle p l lions the concentration compactness principle in the calculus of variations the locally compact case part i and ii ann inst h poincare anal non lineaire 1 1984 109 145 223 283 combined with test functions connected with optimal trudinger moser inequality

stable enrichment and local preconditioning in the particle partition of unity method this paper is concerned with the stability and approximation properties of enriched meshfree and generalized finite element methods in particular we focus on the particle partition of unity method ppum yet the presented results hold for any partition of unity based enrichment scheme the goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h version independent of the regularity of the solution hence we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme to this end we enrich our ppum function space in an enrichment zone hierarchically near the singularities of the solution this initial enrichment however can lead to a severe ill conditioning and can compromise the stability of the discretization to overcome the ill conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and well conditioned basis independent of the employed initial enrichment the construction of this preconditioner is of linear complexity with respect to the number of discretization points we obtain optimal error bounds for an enriched ppum discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we observe a kind of super convergence the results of our numerical experiments clearly show that our enriched ppum with local preconditioning recovers the optimal convergence rate of o h p of the uniform h version globally for the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of o h p with delta geq frac 1 2 for p 1 the convergence rate of our multilevel solver is essentially the same for a purely polynomial approximation and an enriched approximation

note optimal work rest scheduling with exponential work rate decay this note develops optimal multiple rest break models for the case when the decay in work rate is an exponential function of time worked and recovery of work rate potential during rest is a linear function of time rested while empirical evidence indicates that work rate decay functions tend to be best approximated by either exponential or linear functions previous multiple rest break models were based upon a linear work rate decay function efficient solution procedures are developed which require only the solution of a transcendental equation using newton s or an equivalent method although linear linear and exponential linear models are demonstrated to share some important general characteristics a preliminary analysis indicated that use of linear linear policies resulted in substantial sacrifices in productive output when relatively high rates of exponential decay were present the observed losses were exacerbated by higher rates of recovery

a particle partition of unity method part iii a multilevel solver in this sequel to part i siam j sci comput 22 2000 pp 853 890 and part ii siam j sci comput 23 2002 pp 1655 1682 we focus on the efficient solution of the linear block systems arising from a galerkin discretization of an elliptic partial differential equation of second order with the partition of unity method pum we present a cheap multilevel solver for partition of unity pu discretizations of any order the shape functions of a pum are products of piecewise rational pu functions varphi i with supp varphi i omega i and higher order local approximation functions psi i n usually a local polynomial of degree leq p i furthermore they are noninterpolatory in a multilevel approach we have to cope with not only noninterpolatory basis functions but also with a sequence of nonnested spaces due to the meshfree construction hence injection or interpolatory interlevel transfer operators are not available for our multilevel pum therefore the remaining natural choice for the prolongation operators are l2 projections here we exploit the pum construction of the function spaces and a hierarchical construction of the pu itself to localize the corresponding projection problem this significantly reduces the computational costs associated with the setup and the application of the interlevel transfer operators the second main ingredient of our multilevel solver is the use of a block smoother to treat the local approximation functions psi i n for all n simultaneously the results of our numerical experiments in two and three dimensions show that the convergence rate of the proposed multilevel solver is independent of the number of patches card omega i the convergence rate is slightly dependent on the local approximation orders pi

the analysis of multigrid algorithms for pseudodifferential operators of order minus one multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudodifferential operators of order minus one classical multigrid theory deals with the case of differential operators of positive order the pseudodifferential operator gives rise to a coercive form on h sup minus 1 2 omega effective multigrid algorithms are developed for this problem these algorithms are novel in that they use the inner product on h sup minus 1 omega as a base inner product for the multigrid development the authors show that the resulting rate of iterative convergence can at worst depend linearly on the number of levels in these novel multigrid algorithms in addition it is shown that the convergence rate is independent of the number of levels and unknowns in the case of a pseudodifferential operator defined by a single layer potential finally the results of numerical experiments illustrating the theory are presented 19 refs 1 fig 2 tabs

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

maximization of labor productivity through optimal rest break schedules this paper focuses upon the maximization of labor productivity through selection of the optimal number duration and placement of rest breaks it is demonstrated that resolution of the limited validity associated with previous models leads to a mixed integer quadratic programming formulation of this problem efficient analytic procedures are developed which allow solutions to be obtained in minutes using a hand held calculator then the characteristics properties and potential effectiveness of optimal solutions are examined through an illustration from an existing work environment issues relevant to the implementation of the models are presented in a summary of an initial experiment in which one of the models in this paper was used to select rest break policy the results of this experiment indicated that the output achieved using the model s policy was 13 higher than the output achieved when the subjects used their own rest break policies several future research topics are also identified

two level hierarchical fem method for modeling passive microwave devices in recent years multigrid methods have been proven to be very efficient for solving large systems of linear equations resulting from the discretization of positive definite differential equations by either the finite difference method or theh version of the finite element method in this paper an iterative method of the multiple level type is proposed for solving systems of algebraic equations which arise from thep version of the finite element analysis applied to indefinite problems a two levelv cycle algorithm has been implemented and studied with a gauss seidel iterative scheme used as a smoother the convergence of the method has been investigated and numerical results for a number of numerical examples are presented

the mixed cell complex partition of unity method we present a galerkin method for solving partial differential equations which is a blend of ideas from particle based methods on the one side and traditional finite element methods on the other side the method is here named mixed cell complex partition of unity method mccpum it can be arbitrarily considered as being based on a set of scattered particles in the domain and on its boundary or on a delaunay cell decomposition of the domain in contrast to the element free galerkin method and other meshless techniques the partition of unity is not constructed on circular or rectangular supports but rather on overlapping polyhedral supports generated from a voronoi delaunay decomposition of space the mixed cell complex this approach inherits most of the advantages of truly meshless schemes while it greatly facilitates the numerical integration of the weak forms required in galerkin approximations the discretization is exclusively governed by the selection of nodes and the approximation orders associated to the nodes locally here legendre polynomials of arbitrary orders are used the mixed cell complex and the corresponding galerkin discretization are explained numerical examples for the poisson problem in two dimensions are presented and the efficiency of the method is discussed

infinitely many solutions to quasilinear schrodinger system in r n in this work we study the existence of multiple solutions to the quasilinear schrodinger system 0 1 p u a x u p 2 u d 1 f u x u v u m 2 u x r n q v b x v q 2 v d 1 f v x u v v m 2 v x r n u w 1 p r n v w 1 q r n where n 3 1 p q n 0 and m d q p r the potential functions a x b x l r n are positive in r n a major point is that we use the technique in chen 2015 to verify the p s conditions and then apply a version of mountain pass lemma to prove the existence of infinitely many solutions of system 0 1

a multi grid method with a priori and a posteriori level choice for the regularization of nonlinear ill posed problems in this paper we study a multi grid method for the numerical solution of nonlinear systems of equations arising from the discretization of ill posed problems where the special eigensystem structure of the underlying operator equation makes it necessary to use special smoothers we provide uniform contraction factor estimates and show that a nested multigrid iteration together with an a priori or a posteriori chosen stopping index defines a regularization method for the ill posed problem i e a stable solution method that converges to an exact solution of the underlying infinite dimensional problem as the data noise level goes to zero with optimal rates under additional regularity conditions

clustered generalized finite element methods for mesh unrefinement non matching and invalid meshes in spite of significant advancements in automatic mesh generation during the past decade the construction of quality finite element discretizations on complex three dimensional domains is still a difficult and time demanding task in this paper the partition of unity framework used in the generalized finite element method gfem is exploited to create a very robust and flexible method capable of using meshes that are unacceptable for the finite element method while retaining its accuracy and computational efficiency this is accomplished not by changing the mesh but instead by clustering groups of nodes and elements the clusters define a modified finite element partition of unity that is constant over part of the clusters this so called clustered partition of unity is then enriched to the desired order using the framework of the gfem the proposed generalized finite element method can correctly and efficiently deal with i elements with negative jacobian ii excessively fine meshes created by automatic mesh generators iii meshes consisting of several sub domains with non matching interfaces under such relaxed requirements for an acceptable mesh and for correctly defined geometries today s automated tetrahedral mesh generators can practically guarantee successful volume meshing that can be entirely hidden from the user a detailed technical discussion of the proposed generalized finite element method with clustering along with numerical experiments and some implementation details are presented copyright 2006 john wiley sons ltd

multilevel particle partition of unity method this paper is concerned with the particle partition of unity method a meshfree generalization of the finite element method we present the fundamental construction principles and abstract approximation properties of the resulting function spaces v pu moreover we discuss the construction of optimal approximation spaces for a reference application in linear elastic fracture mechanics in particular the presented construction not only yields optimal convergence rates globally independently of the regularity of the solution our method shows a super convergence near the singular points of the solution

arbitrarily smooth generalized finite element approximations this paper presents a procedure to build ck k arbitrarily large generalized finite element fe shape functions defined on non structured finite element meshes the functions have the same support as corresponding global fe lagrangian shape functions meshes with both convex and non convex clouds set of elements sharing a vertex node can be used the so called r functions are used to build ck fe based partition of unity functions with non convex support a technique to combine c0 lagrangian fe shape functions with the proposed ck partition of unity is presented the technique allows the use of ck generalized fe shape functions in parts of the computational domain where their high smoothness is required as in the case of problems with distributional boundary conditions and the less computationally demanding c0 generalized fe shape functions elsewhere in the domain a linear elasticity problem with a concentrated moment is solved using the proposed ck generalized fe method higher order distributional boundary conditions can also be handled by the method a detailed convergence analysis is presented for this class of problems as well as for problems in energy space the integrability of the functions using standard gauss legendre rules is also investigated

a point collocation method based on reproducing kernel approximations a reproducing kernel particle method with built in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations the design and implementation of a galerkin based reproducing kernel particle method however faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions in this paper we present a point collocation method based on reproducing kernel approximations we show that in a point collocation approach the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach the point collocation method based on reproducing kernel approximations however requires the calculation of higher order derivatives that would typically not be required in a galerkin method a correction function and reproducing conditions that enable consistency of the point collocation method are derived the point collocation method is shown to be accurate for several one and two dimensional problems and the convergence rate of the point collocation method is addressed copyright 2000 john wiley sons ltd

existence of soliton solutions for a quasilinear schrodinger equation involving critical exponent in rn abstract mountain pass in a suitable orlicz space is employed to prove the existence of soliton solutions for a quasilinear schrodinger equation involving critical exponent in r n these equations contain strongly singular nonlinearities which include derivatives of the second order such equations have been studied as models of several physical phenomena the nonlinearity here corresponds to the superfluid film equation in plasma physics

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

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

bem with linear complexity for the classical boundary integral operators alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as e g galerkin boundary elements including numerical quadrature and panel clustering these representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations low order discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered we will present fully discrete galerkin boundary element methods where the storage amount and the cpu time grow only linearly with respect to the number of unknowns

optimal order multigrid methods for solving exterior boundary value problems the coupling of boundary elements and finite elements combines the advantage of boundary elements for treating domains extended to infinity and that of finite elements in treating the nonhomogeneity of equations and the complexity of domains in the case of the laplacian by taking a circle or a sphere as the artificial coupling boundary it is shown that the corresponding boundary integral equation can be solved without any cost and the coupled system is reduced to a simple finite element system two multigrid methods are proposed to solve this finite element linear system both methods are of optimal order and can be used to solve such finite element equations as efficiently as to solve those arising from interior boundary value problems numerical experiments are included to show the efficiency and advantages of the methods an apparent significance of the methods is that the boundary elements appear neither in the discretization nor in the coding

remarks on multigrid convergence theorems multigrid has become an important iterative method for the solution of discrete elliptic equations however there is much to be done in the theory of convergence proofs at the present time there are two general two level methods for general convergence proofs an algebraic approach and a duality approach while these theories do not give sharp estimates they provide good general rigorous convergence theorems in this note we study the relationship between these theories while the approach and thought process leading to these theories are different the results are essentially the same indeed the basic estimates required by these theories are the same

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

ground state solutions for quasilinear schrodinger systems this paper is concerned with the quasilinear schrodinger systems in rn u a x 1 u 12 u 2 u 2 u 2 v u v b x 1 v 12 v 2 v 2 u v 2v u x 0 v x 0as x where 0 is a parameter 2 2 2 2 and 2 2nn 2 for n 3 2 for n 1 2 is the critical sobolev exponent by using the nehari manifold method and concentration compactness principle in the orlicz space we prove the existence of ground state solution which localize near the potential well int a 1 0 intb 1 0 for large enough

a particle partition of unity method part ii efficient cover construction and reliable integration in this paper we present a meshfree discretization technique based only on a set of irregularly spaced points x i in mathbb r d and the partition of unity approach in this sequel to m griebel and m a schweitzer siam j sci comput 22 2000 pp 853 890 we focus on the cover construction and its interplay with the integration problem arising in a galerkin discretization we present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme here we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency the use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly now the overall computational costs are dominated by the number of the integration cells we present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency in fact the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method yet the presented method is completely independent of a mesh moreover our method is applicable to general domains and allows for the construction of approximations of any order and regularity

multigrid convergence for nonsymmetric indefinite variational problems and one smoothing step we prove that the multigrid method for linear systems arising from nonsymmetric and indefinite elliptic variational problems converges for all sufficiently small h with one smoothing step only previous proofs required that the number of smoothing steps be within certain bounds which were quite difficult to compute our theory is based on sharp energy norm convergence estimates for the symmetric definite case and lower order terms are treated as perturbations the resulting estimates are so sharp that we obtain even the convergence of the v cycle

on the existence of standing wave solutions for a class of quasilinear schrodinger systems abstract in this paper minimax methods in a suitable orlicz space are employed to establish the existence of standing wave solutions for a class of quasilinear schrodinger systems involving subcritical nonlinearities the systems considered here can model an interaction phenomena in plasma physics

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

nonconforming multigrid method for nonsymmetric and indefinite problems abstract the convergence of w cycle multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems using p 1 nonconforming finite elements is proved

convergence of the multigrid v cycle algorithm for second order boundary value problems without full elliptic regularity the multigrid v cycle algorithm using the richardson relaxation scheme as the smoother is studied in this paper for second order elliptic boundary value problems the contraction number of the v cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps without assuming full elliptic regularity as a consequence the v cycle convergence result of braess and hackbusch is generalized to problems without full elliptic regularity

regularity and multigrid analysis for laplace type axisymmetric equations consider axisymmetric equations associated with laplace type operators we establish full regularity estimates in high order kondrat vetype spaces for possible singular solutions due to the non smoothness of the domain and to the singular coefficients in the operator then we show suitable graded meshes can be used in high order finite element methods to achieve the optimal convergence rate even when the solution is singular using these results we further propose multigrid v cycle algorithms solving the system from linear finite element discretizations on optimal graded meshes we prove the multigrid algorithm is a contraction with the contraction number independent of the mesh level numerical tests are provided to verify the theorem

multigrid convergence for a singular perturbation problem abstract so far there has been no analysis of multigrid methods applied to singularly perturbed dirichlet boundary value problems only for periodic boundary conditions does the fourier transformation mode analysis apply and it is not obvious that the convergence results carry over to the dirichlet case since the eigenfunctions are quite different in the two cases in this paper we prove a close relationship between multigrid convergence for the easily analysable case of periodic conditions and the convergence for the dirichlet case

the convergence rate of a multigrid method with gauss seidel relaxation for the poisson equation the numerical solution of the poisson equation is treated by a multigrid method for a uniform grid the convergence rate can be estimated even for the iteration with a v cycle independently of the shape of the domain as long as it is convex and polygonal the smoothing effect of the gaus seidel relaxation is described by a discrete seminorm which is weaker than the energy norm

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

minimizing the makespan with late start penalties added to processing times in a single facility scheduling problem abstract this paper considers the following scheduling problem minimize the makespan in a single facility problem where the processing time of a job consists of a fixed and a variable part the variable part depends on the start time of the job the objective is to minimize makespan two optimizing algorithms based on dynamic programming and branch and bound techniques and five heuristic rules have been proposed the heuristic solutions have been compared with the optimal solution for upto 7 jobs to identify the best heuristic the detailed computational results for the branch and bound and the best heuristic rule are then obtained for problems with up to 15 jobs

a conjugate gradient method and a multigrid algorithm for morley s finite element approximation of the biharmonic equation the numerical solution of the linear equations arising from morley s nonconforming displacement method is studied a suitable preconditioning for the conjugate gradient method is described furthermore the nonconformity of the discretization necessitates some non standard ingredients of multigrid algorithms

scheduling in a contaminated area a model and polynomial algorithms abstract there are n jobs to be scheduled in a contaminated area the jobs can be rescue de activation or cleaning works to be executed by a single worker in an area contaminated with radio active or chemical materials precedence relations can be given on the set of jobs an execution of each job can be preempted however the length of the minimal uninterrupted work period is given and it is the same for all jobs each work period for a job should be accompanied by a rest period whose length depends on the start time of the work period and its length we focus on a short term planning problem we show that this problem can be modelled by a scheduling problem with start time dependent job processing times the dependency functions are exponentially decreasing ones we also construct two polynomial time algorithms for the both cases with and without precedence constraints

convergence analysis of perturbed two grid and multigrid methods we consider multigrid methods for symmetric positive definite linear systems we present a new algebraic convergence analysis of two grid schemes with inexact solution of the coarse grid system this analysis allows us to bound the convergence factor of such perturbed two grid schemes assuming only a certain bound on the convergence factor for the unperturbed scheme with exact solution of the coarse grid system applied to multigrid methods with the standard w cycle this analysis shows that if the convergence factor of the unperturbed two grid method is uniformly bounded by sigma 1 2 then the convergence factor of the multigrid method is uniformly bounded by sigma 1 sigma the analysis is purely algebraic and requires only that pre and postsmoothing are applied in a symmetric way it covers both geometric and algebraic multigrid methods and the coarse grid matrix may be of any type not necessarily galerkin

a multilevel algorithm for the biharmonic problem a finite element discretization of the mixed variable formulation of the biharmonic problem is considered a multilevel algorithm for the numerical solution of the discrete equations is described convergence is proved under the assumption ofh 3 regularity

split level iteration many techniques are used for solving large sparse linear systems some of the most popular methods for solving the more difficult problems combine direct and iterative algorithms to yield efficient multistage procedures although each stage is standard the manner in which stages are combined is often novel both multigrid and preconditioned conjugate gradients fall into this class of numerical methods in lo two promising algorithms were described with supportive numerical studies the first was cyclic application of model problem adi iteration followed by a conjugate gradient iteration on the deviation of the model problem from the actual problem some new developments relating to this procedure are described in a companion paper in this journal 121 the second was a two level iteration in which a variational correction with a coarse grid basis was applied cyclically to the result of conventional iteration over a fine grid this was the first published implementation of a multigrid method based on the variational techniques that provide the foundations for finite element computation in these initial two level variational studies certain advantages of additive over multiplicative coarse mesh correction were not recognized the coefficient matrix for additive correction does not depend on the fine mesh iterate and need thus be computed only once moreover it need only be factored once for direct solution the right hand side of the correction equations varies each cycle brandt 4 and others have exploited this in multigrid and related methods some numerical comparisons of additive and multiplicative correction for two level iteration are given in 111 also it was not recognized that under suitable conditions one may use the two level iteration as a preconditioner for conjugate gradients thus combining the two initially distinct methods as suggested subsequently in 1 7 9 some insight into the nature of such an iteration may be gained by a qualitative analysis of error behavior the fine mesh iteration acts primarily on high mcde error components and the coarse mesh correction reduces low mode components in some implementations the two iteration matrices commute but in most they do not in the commuting case careful choice of iteration parameters precludes benefits from the conjugate gradient third stage in the noncommuting case the conjugate gradient stage should act on error persisting because of component mixing between the other two stages the automatic extrapolation in the conjugate gradient method greatly reduces convergence sensitivity to choice of iteration parameters for the fine mesh iteration

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

a posteriori error estimation for generalized finite element methods in this paper we address the problem of a posteriori error estimation for generalized finite element methods based on the partition of unity method the computational results focus on the question of the reliability of the error estimators and its assessment

implementation of boundary conditions for meshless methods abstract boundaries and boundary conditions are an aspect of the numerical solution of partial differential equations where meshless methods have had to surmount many initial difficulties due to the lack of a finite element like kronecker delta condition furthermore it is frequently desirable especially in fluid mechanics to impose general nonlinear boundary and interface constraints this paper describes a computationally efficient algorithm based on d alembert s principle that can be used for general constraints both in meshless methods and finite elements first a method of partitioning meshless shape functions suitable for imposing linear boundary conditions is developed subsequently an analogous method is developed for nonlinear constraints special attention is given to imposing general boundary and fluid structure interface conditions on the navier stokes equations in terms of conservative variables numerical results using d alembert s principle with the reproducing kernel particle method rkpm including viscous supersonic flow past a naca 7012 airfoil are shown

wasserstein space over the wiener space the goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the wiener space based on and extending fundamental results of feyel ustunel carrying out the program of ambrosio gigli savare we present a complete characterization of the derivative processes for certain class of absolutely continuous curves we prove existence of the gradient flow curves for the relative entropy w r t the wiener measure and identify these gradient flow curves with solutions of the ornstein uhlenbeck evolution equation

solutions for a quasilinear schrodinger equation a dual approach abstract we consider quasilinear stationary schrodinger equations of the form 1 u u 2 u g x u x r n introducing a change of unknown we transform the search of solutions u x of 1 into the search of solutions v x of the semilinear equation 2 v 1 1 2f 2 v g x f v x r n where f is suitably chosen if v is a classical solution of 2 then u f v is a classical solution of 1 variational methods are then used to obtain various existence results

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

an adaptive hp version of the multilevel particle partition of unity method abstract this paper is concerned with the hp adaptive multilevel solution of second order elliptic partial differential equations using the meshfree particle partition of unity method the proposed refinement scheme automatically constructs new discretization points or particles the meshfree analogue of an adaptive h refinement and local approximation spaces with better local resolution a p refinement the refinement process is steered with the help of an a posteriori subdomain type error indicator we present results of numerical experiments in two and three space dimensions which demonstrated the overall efficiency of the proposed scheme

generalized finite element method for second order elliptic operators with dirichlet boundary conditions we introduce a method for approximating essential boundary conditions conditions of dirichlet type within the generalized finite element method gfem framework our results apply to general elliptic boundary value problems of the form email protected i j 1 n a i ju x i x email protected i 1 nb iu x i cu f in w u 0 on email protected where w is a smooth bounded domain as test trial spaces we consider sequences of gfem spaces s m m 1 which are nonconforming that is s email protected h 0 1 w we assume that email protected l 2 w

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

an algebraic treatment of essential boundary conditions in the particle partition of unity method this paper is concerned with the treatment of essential boundary conditions in meshfree methods in particular we focus on the particle partition of unity method ppum however the proposed technique is applicable to any partition of unity based approach we present an efficient scheme for the automatic construction of a direct splitting of a ppum function space into the degrees of freedom suitable for the approximation of the dirichlet data and the degrees of freedom that remain for the approximation of the pde by simple linear algebra notably our approach requires no restrictions on the distribution of the discretization points nor on the employed local approximation spaces we attain the splitting of the global function space from the respective direct splittings of the employed local approximation spaces hence the global splitting can be computed with sub linear complexity due to this direct splitting of the meshfree ppum function space we can implement a conforming local treatment of essential boundary data so that the realization of dirichlet boundary values in the meshfree ppum is straightforward the presented approach yields an optimally convergent scheme which is demonstrated by the presented numerical results

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

iterative solution methods for plate bending problems multigrid and preconditioned cg algorithm the numerical solution of the linear equations arising from a discretization of the plate bending problem based on the zienkiewicz triangle is studied the finite element scheme proceeds from the mindlin reissner formulation with modified shear energy however the kirchhoff condition is imposed on discrete points a multigrid algorithm is analyzed and numerical examples are provided furthermore a suitable preconditioning is established for the use of conjugate gradients

overcoming corner singularities using multigrid methods we consider the poisson equation delta u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain o we develop a finite element multigrid method on quasi uniform grids that obtains cal o h m 1 epsilon convergence in the h 1 o norm for any positive epsilon when f in h m o the cost of the method is proportional to the number of elements in the triangulation the results of this paper can be generalized to other equations and other boundary conditions

a class of difference schemes with flexible local approximation solutions of many physical problems have salient local features that are qualitatively known a priori for example singularities at point sources edge and corners boundary layers derivative jumps at material interfaces strong dipole field components near polarized spherical particles cusps of electronic wavefunctions at the nuclei electrostatic double layers around colloidal particles etc the known methods capable of providing flexible local approximation of such features include the generalized finite element partition of unity method special variational difference schemes in broken sobolev spaces and a few other specialized techniques in the proposed new class of flexible local approximation methods flame a desirable set of local approximating functions such as cylindrical or spherical harmonics plane waves harmonic polynomials etc defines a finite difference scheme on a chosen grid stencil one motivation is to minimize the notorious staircase effect at curved and slanted interface boundaries however the new approach has much broader applications as illustrative examples the paper presents arbitrarily high order 3 point schemes for the 1d schrodinger equation and a 1d singular equation schemes for electrostatic interactions of colloidal particles electromagnetic wave propagation and scattering plasmon resonances moreover many classical finite difference schemes including the collatz mehrstellen schemes are direct particular cases of flame

preemptive scheduling with simple linear deterioration on a single machine in this paper we study the problem of scheduling n deteriorating jobs with release dates on a single machine the processing time of a job is assumed to be the product of its deteriorating rate and its starting time precedence relations may be imposed on the set of jobs unlike the classical time dependent scheduling problems we assume that the execution of a job can be preempted in the sense that the job s deteriorating rate is reduced when it is preempted and each continuously processed part of a job can be regarded as an independent job with a specified deteriorating rate the objective is to minimize some common regular scheduling performance measures we first show that minimizing a class of regular symmetric functions is polynomially solvable then we construct an o n 2 algorithm for minimizing the maximum job completion cost with or without precedence constraints finally we show that minimizing the total weighted completion time is np hard even if there are only two distinct release dates

multigrid methods for nonconforming finite element methods the present paper analyzes a multigrid algorithm for the crouzeix raviart discretization of the poisson and stokes equations in two and three dimensions the central point is the construction of easily computable l 2 projections based on suitable quadrature rules for the transfer from coarse to fine grids and vice versa