a multigrid method for the mortar type morley element approximation of a plate bending problem in this paper a multigrid algorithm is presented for the mortar element method for the morley nonconforming plate element based on the multigrid theory developed by bramble pasciak and xu in math comp 56 1991 pp 1 34 we prove that the w cycle multigrid method is optimal i e the convergence rate is independent of the mesh size and mesh level moreover a variable v cycle multigrid preconditioner is constructed which results in a preconditioned system with a uniformly bounded condition number

appending boundary conditions by lagrange multipliers analysis of the lbb condition this paper is concerned with the analysis of discretization schemes for second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of lagrange multipliers specifically we show how the validity of the ladyskaja babusska brezzi lbb condition for the corresponding saddle point problems depends on the various ingredients of the involved discretizations the main result states that the lbb condition is satisfied whenever the discretization step length on the boundary h gamma sim 2 ell is somewhat bigger than the one on the domain h omega sim 2 j this is quantified through constants stemming from the trace theorem norm equivalences for the multiplier spaces on the boundary and direct and inverse inequalities in order to better understand the interplay of these constants we then specialize the setting to wavelet discretizations in this case the stability criteria can be stated solely in terms of spectral properties of wavelet representations of the trace operator we conclude by illustrating our theoretical findings by some numerical experiments we stress that the results presented here apply to any spatial dimension and to a wide selection of lagrange multiplier spaces which in particular need not be traces of the trial spaces however we do always assume that a hierarchy of nested trial spaces is given

multigrid on the interface for mortar mixed finite element methods for elliptic problems we consider mixed finite element approximations of second order elliptic equations on domains that can be described as a union of subdomains or blocks we assume that the subdomain grids are locally defined and need not match across the block boundaries specially chosen mortar finite element spaces are introduced on the interfaces for approximating the scalar variable pressure the mortars also serve as lagrange multipliers for imposing flux matching conditions the method is implemented by reducing the algebraic system to a positive definite interface problem in the mortar spaces this problem is then solved using a multigrid on the interface with conjugate gradient smoothing the algorithm is very efficient in a distributed parallel computing environment as only subdomain solves are required on each conjugate gradient iteration the standard variational assumptions for the multigrid are not satisfied since the interface bilinear forms vary from level to level we present theoretical results for the convergence of the v cycle and the w cycle computational results in two and three dimensions are given to illustrate and confirm the theory

a cascadic multigrid algorithm for the stokes equations a variant of multigrid schemes for the stokes problem is discussed in particular we propose and analyse a cascadic version for the stokes problem the analysis of the transfer between the grids requires special care in order to establish that the complexity is the same as that for classical multigrid algorithms

a computational multilevel approach for solving 2d navier stokes equations over non matching grids a multilevel approach with parallel implementation is developed for obtaining fast solutions of the navier stokes equations solved on domains with non matching grids the method relies on computing solutions over different subdomains with different multigrid levels by using multiple processors a local vanka type relaxation operator for the multigrid solution of the navier stokes system allows solutions to be computed at the element level the natural implementation on a multiprocessor architecture results in a straightforward and flexible algorithm numerical computations are presented using benchmark applications in order to support the method parallelization is discussed to achieve proper accuracy load balancing and computational efficiency between different processors 2005 elsevier b v all rights reserved

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

substructuring preconditioners for the q_1 mortar element method the q 1 mortar element method is a non conforming finite element method with q 1 elements based on domain decomposition for the laplace equation it yields an ill conditioned linear system for solving the linear system the so called preconditioned conjugate gradient method in a subspace is used preconditioners are proposed and estimates on condition numbers and arithmetical complexity are given finally numerical experiments are presented

bv estimates for mortar methods in linear elasticity this paper is concerned with the convergence of mortar methods applied to linear elasticity we prove that the conventional mesh dependent norms used in the analysis of mortar methods are bounded below by the bv norm when combined with standard results this bound establishes a decomposition independent and mesh independent proof of the convergence of mortar methods in linear elasticity

hierarchical a posteriori error estimators for mortar finite element methods with lagrange multipliers hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods a weak continuity condition at the interfaces is enforced by means of lagrange multipliers the two proposed error estimators are based on a defect correction in higher order finite element spaces and an adequate hierarchical two level splitting the first provides upper and lower bounds for the discrete energy norm of the mortar finite element solution whereas the second also estimates the error for the lagrange multiplier it is shown that an appropriate measure for the nonconformity of the mortar finite element solution is the weighted l2 norm of the jumps across the interfaces

multiplier spaces for the mortar finite element method in three dimensions we consider the construction of multiplier spaces for use with the mortar finite element method in three spatial dimensions on globally or locally quasi uniform meshes a set of abstract conditions is given for the multiplier spaces which are sufficient to guarantee a stable and convergent mortar approximation three examples of multipliers satisfying these conditions are presented the first one is a dual basis example while the remaining two are based on finite volumes finally the results of computational examples illustrating the theory are reported

a cal v cycle multigrid approach for mortar finite elements mortar methods based on dual lagrange multipliers provide a flexible tool for the numerical approximation of partial differential equations the associated finite element spaces are in general nonconforming and nonnested optimal multigrid results have previously been established for cal w cycle and the variable cal v cycle multigrid methods in this paper we introduce a new multigrid method based on a nested sequence of modified mortar spaces for which we can establish that the cal v cycle with one smoothing step has contraction numbers uniformly bounded away from one to obtain nested mortar spaces we apply a product form of certain corrections at the interfaces numerical results demonstrate the efficiency of the resulting multigrid solver

on a generalized l_2 projection and some related stability estimates in sobolev spaces in this paper we investigate a stability estimate needed in hybrid finite and boundary element methods especially in hybrid coupled domain decomposition methods including mortar finite elements this stability estimate is equivalent to the stability of a generalized l 2 projection in certain sobolev spaces using piecewise linear trial spaces and appropriate piecewise constant test spaces the stability of the generalized l 2 projection is proved assuming some mesh conditions locally

a stabilized domain decomposition method with nonmatching grids for the stokes problem in three dimensions we present and study a nonconforming domain decomposition method for the discretization of the three dimensional stokes problem in the velocity pressure formulation the approximation is based on some local mixed finite elements for nonmatching tetrahedral grids the aim pursued is a systematic construction of the mortared discrete velocity space the pressure being not subjected to any matching constraints across the interfaces using the bubble stabilization techniques applied in brezzi and marini s paper to the three fields method math comp 70 2001 pp 911 934 allows us to define an algorithm which is easy to implement the numerical analysis relies on the pressure splitting argument of boland and nicolaides and allows us to establish an inf sup condition with a constant that does not depend on the mesh size or on the total number of the subdomains then by the berger scott strang lemma written down for our saddle point system we derive optimal accuracy results

multigrid for the mortar finite element method a multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed these problems are posed on domains partitioned into subdomains each of which is independently triangulated in a multilevel fashion the multilevel mortar finite element spaces based on such triangulations which need not align across subdomain interfaces are in general not nested suitable grid transfer operators and smoothers are developed which lead to a variable v cycle preconditioner resulting in a uniformly preconditioned algebraic system computational results illustrating the theory are also presented

exotic bailey slater spt functions i group a abstract we introduce several spt type functions that arise from bailey pairs we prove simple ramanujan type congruences for these functions which can be explained by a spt crank type function the spt crank type functions are constructed by adding an extra variable z into the generating functions we find dissections when z is a certain root of unity as has been done for many rank and crank difference formulas of various partition type objects our formulas require an identity of s h chan on generalized lambert series

a non conforming computational methodology for modeling coupled problems a variety of engineering applications that involve coupling different physical phenomena often require detailed finite element analysis to be carried out over complex domains often such analysis may be accomplished by dividing the global domain into several local subdomains over each of which a local model can be analyzed independently the global solution can then be constructed by suitably piecing together local solutions from these individually modeled subdomains however during the assembly it is often too cumbersome or even infeasible to coordinate the meshes over separate subdomains one must therefore employ non conforming techniques to accomplish such modeling in this paper we develop a non conforming computational methodology via the mortar finite element method to solve a coupled problem where we are interested in determining the effects of temperature variations on a given flow or the transfer of heat within the flow using this method the solution over different subdomains with different multigrid levels is efficiently computed our numerical results clearly suggest that the proposed methodology is robust and stable 2005 elsevier ltd all rights reserved

on the rank and the crank modulo 4 and 8 in this paper we prove some identities conjectured by lewis for the rank and crank of partitions concerning the modulo 4 and 8 these identities are similar to dyson s identities for the rank modulo 5 and 7 which give a combinatorial interpretation to ramanujan s partition congruences for this we use multisection of series and some of the results that watson established for the third order mock theta functions

a numerical study of feti algorithms for mortar finite element methods the finite element tearing and interconnecting feti method is an iterative substructuring method using lagrange multipliers to enforce the continuity of the finite element solution across the subdomain interface mortar finite elements are nonconforming finite elements that allow for a geometrically nonconforming decomposition of the computational domain into subregions and at the same time for the optimal coupling of different variational approximations in different subregions we present a numerical study of feti algorithms for elliptic self adjoint equations discretized by mortar finite elements several preconditioners which have been successful for the case of conforming finite elements are considered we compare the performance of our algorithms when applied to classical mortar elements and to a new family of biorthogonal mortar elements and we discuss the differences between enforcing mortar conditions instead of continuity conditions for the case of matching nodes across the interface our experiments are carried out for both two and three dimensional problems and include a study of the relative costs of applying different preconditioners for mortar elements

simulation of a magneto mechanical damping machine analysis discretization results this paper presents and analyzes a method for the simulation of the dynamical behavior of a coupled magnetomechanical system such as a damping machine we consider a two dimensional model based on the transverse magnetic formulation of the eddy currents problem for the electromagnetic part and on the motion equation of a rotating rigid body for the mechanical part the magnetic system is discretized in space by means of lagrangian finite elements and the sliding mesh mortar method is used to account for the rotation in time a one step euler method is used implicit for the magnetic and velocity equations the coupled differential system is solved with an explicit procedure 2002 elsevier science b v all rights reserved

higher order mortar finite element methods in 3d with dual lagrange multiplier bases mortar methods with dual lagrange multiplier bases provide a flexible efficient and optimal way to couple different discretization schemes or nonmatching triangulations here we generalize the concept of dual lagrange multiplier bases by relaxing the condition that the trace space of the approximation space at the slave side with zero boundary condition on the interface and the lagrange multiplier space have the same dimension we provide a new theoretical framework within this relaxed setting which opens a new and simpler way to construct dual lagrange multiplier bases for higher order finite element spaces as examples we consider quadratic and cubic tetrahedral elements and quadratic serendipity hexahedral elements numerical results illustrate the performance of our approach

isogeometric analysis for csg models abstract the iga isogeometric analysis was developed as a bridge between cad and cae with its highly precise geometry representation however iga can only do little with the models that cad systems generated in csg constructive solid geometry manner because the trimmed domain exists this paper presents a method named constructive solid isogeometric analysis csiga to work for csg models that are composed of primitive solids with union subtraction and intersection operators first this work decomposes the csg models into several primitive solids represented by nurbs separately then cuts the primitives with their adjacent ones to produce a set of trimmed tri variable nurbs primitives and finally uses boolean operators to compose them as to the single trimmed solid an integration scheme is presented meanwhile a combination scheme has been developed to deal with the union of multiple trimmed solids by using a mortar method the formulation is verified by applying the isogeometric analysis on several solids which have theoretical solutions such as elliptic cylinder and combined brick and some tests on multiple trimmed solids are done too it is shown that the csiga method can deal with csg models efficiently

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

on the continuity of the srb entropy for endomorphisms we consider classes of dynamical systems admitting markov induced maps under general assumptions which in particular guarantee the existence of srb measures we prove that the entropy of the srb measure varies continuously with the dynamics we apply our result to a vast class of non uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the srb measure in particular we show that the srb entropy of viana maps varies continuously with the map

isogeometric mortar methods the application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically for the lagrange multiplier two choices of uniformly stable spaces are presented both of them are spline spaces but of a different degree in one case we consider an equal order pairing for which a cross point modification based on a local degree reduction is required in the other case the degree of the dual space is reduced by two compared to the primal this pairing is proven to be inf sup stable without any necessary cross point modification several numerical examples confirm the theoretical results and illustrate additional aspects

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

additive schwarz methods for elliptic mortar finite element problems two variants of the additive schwarz method for solving linear systems arising from the mortar finite element discretization on nonmatching meshes of second order elliptic problems with discontinuous coefficients are designed and analyzed the methods are defined on subdomains without overlap and they use special coarse spaces resulting in algorithms that are well suited for parallel computation the condition number estimate for the preconditioned system in each method is proportional to the ratio h h where h and h are the mesh sizes and it is independent of discontinuous jumps of the coefficients for one of the methods presented the choice of the mortar nonmortar side is independent of the coefficients

on the seventh order mock theta functions in a recent paper h we proved the mock theta conjectures these are identities stated by ramanujan in his lost notebook jr2 pp 19 20 involving two of the 5th order mock 0 functions in fact ramanujan gave one such identity for each of the ten 5th order functions as shown in a g1 these ten identities are all equivalent to the two proved in i hi in this paper we prove the corresponding identities for ramanujan s 7th order mock 0 functions and show how they are related to rank differences of partitions to state our results we first introduce some standard notation if q and x are complex numbers with ql 1 and n is a nonnegative integer then

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

srb measures and pesin s entropy formula for endomorphisms a celebrated result conjectured by ruelle and proved by ledrappier strelcyn and young in the smooth ergodic theory of diffeomorphisms is that an invariant measure of a c 2 diffeomorphism satisfies pesin s entropy formula if and only if it has absolutely continuous conditional measures on unstable manifolds 41 42 the later property is known as srb property of the invariant measure this result was also successfully generalized to random dynamical systems of diffeomorphisms 51

the crouzeix raviart fe on nonmatching grids with an approximate mortar condition a new approximate mortar condition is proposed for the lowest order crouzeix raviart finite element on nonmatching grids which uses only the nodal values on the interface for the calculation of the mortar projection this approach allows for improved and more flexible algorithms compared to those for the standard mortar condition where nodal values in the interior of a sub domain those closest to a mortar side of the subdomain are also required in the calculation

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

inequalities for ranks of partitions and the first moment of ranks and cranks of partitions abstract we prove two monotonicity properties of n m n the number of partitions of n with rank m they are i for any nonnegative integers m and n n m n n m 2 n and ii for any nonnegative integers m and n such that n 12 n m 2 n m n n m n 1 g e andrews b kim and the first author introduced ospt n a function counting the difference between the first positive rank and crank moments they proved that ospt n 0 in another article k bringmann and k mahlburg gave an asymptotic estimate for ospt n the two monotonicity properties for n m n lead to stronger inequalities for ospt n that imply the asymptotic estimate

multigrid for the mortar element method for p1 nonconforming element in this paper a multigrid algorithm is presented for the mortar element method for p1 nonconforming element based on the theory developed by bramble pasciak xu in 5 we prove that the w cycle multigrid is optimal i e the convergence rate is independent of the mesh size and mesh level meanwhile a variable v cycle multigrid preconditioner is constructed which results in a preconditioned system with uniformly bounded condition number

domain decomposition methods for mortar finite element discretizations of plate problems mortar finite element methods for clamped plate problems are discussed in the subdomains conforming the reduced hsieh clough tocher hct macroelement defined on nonmatching triangulations is utilized two additive schwarz methods for solving the system of linear equations arising from this discretization are designed and analyzed the condition numbers of the preconditioned systems are almost optimal i e they are independent of the number of subdomains and grow only logarithmically with respect to the mesh parameters

a multigrid algorithm for the mortar finite element method the objective of this paper is to develop and analyze a multigrid algorithm for the system of equations arising from the mortar finite element discretization of second order elliptic boundary value problems in order to establish the inf sup condition for the saddle point formulation and to motivate the subsequent treatment of the discretizations we first revisit briefly the theoretical concept of the mortar finite element method employing suitable mesh dependent norms we verify the validity of the ladyzhenskaya babuska brezzi lbb condition for the resulting mixed method and prove an l2 error estimate this is the key for establishing a suitable approximation property for our multigrid convergence proof via a duality argument in fact we are able to verify optimal multigrid efficiency based on a smoother which is applied to the whole coupled system of equations we conclude with several numerical tests of the proposed scheme which confirm the theoretical results and show the efficiency and the robustness of the method even in situations not covered by the theory

inverse inequalities on non quasi uniform meshes and application to the mortar element method we present a range of mesh dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape regular but possibly non quasi uniform meshes these inequalities involve norms of the form h u w s p for positive and negative s and where h is a function which reflects the local mesh diameter in an appropriate way the only global parameter involved is n the total number of degrees of freedom in the finite element space and we avoid estimates involving either the global maximum or minimum mesh diameter our inequalities include new variants of inverse inequalities as well as trace and extension theorems they can be used in several areas of finite element analysis to extend results previously known only for quasi uniform meshes to the locally refined case here we describe applications to i the theory of nonlinear approximation and ii the stability of the mortar element method for locally refined meshes

the mortar finite element method for 3d maxwell equations first results in the framework of domain decomposition methods we extend the main ideas of the mortar element method to the numerical solution of maxwell s equations in wave form by h curl conforming finite elements the method we propose turns out to be a new nonconforming nonoverlapping domain decomposition method where nonmatching grids are allowed at the interfaces between subdomains a model problem is considered the convergence of the discrete approximation is analyzed and an error estimate is provided the method is proven to be slightly suboptimal with a loss of a factor scriptstyle sqrt rm ln h with respect to the degree of polynomials in order to achieve this convergence result we nevertheless need extra regularity assumptions on the solution of the continuous problem

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

mortar boundary elements we establish a mortar boundary element scheme for hypersingular boundary integral equations representing elliptic boundary value problems in three dimensions we prove almost quasi optimal convergence of the scheme in broken sobolev norms of order 1 2 subdomain decompositions can be geometrically nonconforming and meshes must be quasi uniform only on subdomains numerical results confirm the theory

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

uniform hp convergence results for the mortar finite element method the mortar finite element is an example of a non conforming method which can be used to decompose and re compose a domain into sub domains without requiring compatibility between the meshes on the separate components we obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used without assuming quasiuniformity for the meshes our results establish that the method is optimal when non quasiuniform h or hp methods are used such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain we also give an error estimate for when the p version is used numerical results for hp and hp mortar fems show that these methods behave as well as conforming fems an hp extension theorem is also proved

duality estimates and multigrid analysis for saddle point problems arising from mortar discretizations we present an abstract framework for the analysis of multigrid methods for a saddle point problem arising from mortar finite element discretizations in contrast to other approaches the iterates do not have to be in the positive definite subspace moreover our approach also covers the case of nonnested lagrange multiplier spaces we apply the multigrid method to different mortar settings including dual lagrange multipliers linear elasticity and a rotating geometry numerical results in two dimensions and three dimensions demonstrate the flexibility efficiency and reliability of our multigrid method

a multigrid method based on the unconstrained product space for mortar finite element discretizations the mortar finite element method allows the coupling of different discretizations across subregion boundaries in the original mortar approach the lagrange multiplier space enforcing a weak continuity condition at the interfaces is defined as a modified finite element trace space here we present a new approach where the lagrange multiplier space is replaced by a dual space without losing the optimality of the a priori bounds we introduce new dual spaces in 2d and 3d using the biorthogonality between the nodal basis functions of this lagrange multiplier space and a finite element trace space we derive an equivalent symmetric positive definite variational problem defined on the unconstrained product space the introduction of this formulation is based on a local elimination process for the lagrange multiplier this equivalent approach is the starting point for the efficient iterative solution by a multigrid method to obtain level independent convergence rates for the cw cycle we have to define suitable level dependent bilinear forms and transfer operators numerical results illustrate the performance of our multigrid method in 2d and 3d

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