comparison results for quasilinear elliptic equations via a picone type identity part i quasilinear cases abstract using some identities of picone s types comparison results and uniqueness theorems are established for some general quasilinear elliptic equations

second order accuracy of brenier s time discrete method for nonlinear systems of conservation laws an approximation tilde u x t to the smooth solution of a hyperbolic system of conservation laws is shown to be second order accurate in time i e tilde u x t u x t o t 3 this approximation is based on linearized characteristics and is related to various other perturbation techniques for hyperbolic systems proving second order accuracy is of interest because certain fully discrete numerical methods can be viewed as spatial discretization of the time discrete method

a feti dp formulation of three dimensional elasticity problems with mortar discretization in this paper a feti dp formulation for three dimensional elasticity on nonmatching grids over geometrically nonconforming subdomain partitions is considered to resolve the nonconformity of the finite elements a mortar matching condition is imposed on the subdomain interfaces faces a feti dp algorithm is then built by enforcing the mortar matching condition in dual and primal ways in order to make the feti dp algorithm scalable a set of primal constraints which include average and momentum constraints over interfaces are selected from the mortar matching condition a condition number bound c 1 log h h 2 is then proved for the feti dp formulation for the elasticity problems with discontinuous material parameters only some faces need to be chosen as primal faces on which the average and momentum constraints are imposed numerical results are included

a primal formulation for imposing periodic boundary conditions on conforming and nonconforming meshes abstract a variational multiscale discontinuous galerkin vmdg method is developed for microscale modeling of domains containing conforming and non conforming meshes essentially the product of the applied volume average strain or macro strain and the domain diameter acts as an imposed displacement jump within the vmdg terms hence the method is suitable for modeling deformation of both block and truly self periodic representative volume elements rves the primal displacement field and macro strain are the only unknowns because the method eliminates the lagrange multiplier lm enforcement of the kinematic constraint rigorous derivation of the method provides a framework to accommodate either the macro stress or macro strain as the driver of the microscale boundary value problem the method is developed first for finite deformations and then specialized to small deformation kinematics algorithmic modifications to the method are also studied for their effects on tangent symmetry and convergence rate the results from numerical studies for isotropic and anisotropic materials show that the proposed method is robust accurate stable and variationally consistent for modeling complicated conforming and nonconforming rves

on shape preserving quadratic spline interpolation in this paper we discuss the design of algorithms for interpolating discrete data using c 1 quadratic splines in such a way that the monotonicity and or convexity of the data is preserved the analysis culminates in an interactive algorithm which takes full advantage of the flexibility which quadratic splines permit

efficient solvers for saddle point problems arising from domain decompositions with lagrange multipliers in this paper we are concerned with the domain decomposition method with lagrange multipliers for solving three dimensional elliptic problems with variable coefficients we shall first introduce a weighted saddle point problem resulting from this domain decomposition which can be solved by existing iterative methods then we will construct two simple preconditioners one for the system associated with the displacement variable and the other for the schur complement system associated with the multiplier variable that are applicable to various discretization schemes the new preconditioners possess such local properties that they can be implemented cheaply we will show that the condition number of the global preconditioned system grows only as the logarithm of the dimension of the local problem associated with an individual substructure and is independent of the large variations of the coefficients across the local interfaces

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

incorporation of linear multipoint constraints in domain decomposition based iterative solvers part ii blending feti dp and mortar methods and assembling floating substructures two approaches are presented for extending the feti dp domain decomposition based iterative method to the solution of problems with linear multipoint constraints lmpcs both cases of arbitrary lmpcs and lmpcs generated by the mortar method for enforcing a weak continuity of the solution across non matching finite element interfaces are considered in the first approach the lmpcs are addressed during the preconditioning step but enforced only at convergence in the second approach the lmpcs are enforced at each iteration through the solution of an auxiliary coarse problem both extended feti dp solvers are benchmarked on a parallel processor for a wide range of constrained structural mechanics problems the obtained results reveal that both solvers are numerically scalable and complement each other from the memory and cpu performance viewpoints

finite deformation frictional mortar contact using a semi smooth newton method with consistent linearization a two dimensional finite deformation frictional contact formulation with coulomb s law is presented the approach considers multibody contact and is based on a mortar formulation the enforcement of contact constraints is realized with dual lagrange multipliers these alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase friction kinematic variables are formulated in an objective way and enter non smooth complementarity functions for expressing the contact constraints an active set strategy is derived by applying a semismooth newton method which treats contact nonlinearities material and geometrical nonlinearities in one single iterative scheme by further carrying out a consistent linearization for both normal and frictional contact forces and constraints a robust and highly efficient algorithm for linear and higher order quadratic interpolation is achieved efficiency of the proposed method and quality of results are demonstrated in serveral exemples

existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations in connection with the maximizing problem for the functional r u u lq u lp in w01 p 0 we consider the equation div u p 2 u x u q 2 u x x 1 p resp q p e has no nontrivial resp nonnegative nontrivial solution belonging to the class p u w01 p lq xi u q 2u lp p 1 i 1 2 n w01 p provided that is star shaped the crucial point of the proof of our result is to obtain an l estimate of weak solutions and to verify a certain pohozaev type inequality for weak solutions belonging to p

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

weighted rational cubic spline interpolation and its application abstract in qi duan et al korean j comput appl math 6 1 1999 203 215 the authors have discussed constrained interpolation problems by means of rational cubic spline interpolation with linear denominators but there are still some cases in which the constrained interpolation cannot be solved in this paper the weighted rational cubic spline interpolation has been constructed using the rational cubic spline with linear denominator and the rational cubic spline based on function values by these the problems to constrain the weighted rational interpolation curves to lie strictly above or below a given piecewise linear curve and between two given piecewise linear curves can be solved completely also the approximation properties of these weighted rational cubic splines are studied

two level schwarz algorithms with overlapping subregions for mortar finite elements preconditioned conjugate gradient methods based on two level overlapping schwarz methods often perform quite well such a preconditioner combines a coarse space solver with local components which are defined in terms of subregions that form an overlapping covering of the region on which the elliptic problem is defined precise bounds on the rate of convergence of such iterative methods have previously been obtained in the case of conforming lower order and spectral finite elements as well as in a number of other cases in this paper this domain decomposition algorithm and analysis are extended to mortar finite elements it is established that the condition number of the relevant iteration operator is independent of the number of subregions and varies with the relative overlap between neighboring subregions linearly as in the conforming cases previously considered

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

mortar formulation for a class of staggered discontinuous galerkin methods abstract a mortar formulation is developed and analyzed for a class of staggered discontinuous galerkin sdg methods applied to second order elliptic problems in two dimensions the computational domain consists of nonoverlapping subdomains and a triangulation is provided for each subdomain which need not conform across subdomain interfaces this feature allows a more flexible design of discrete models for problems with complicated geometries shocks or singular points a mortar matching condition is enforced on the solutions across the subdomain interfaces by introducing a lagrange multiplier space moreover optimal convergence rates in both l 2 and discrete energy norms are proved numerical results are presented to show the performance of the method

monotonicity preserving c2 rational cubic spline for monotone data abstract designers in industries need to generate splines which can interpolate the data points in such a way that they preserve the inherited shape characteristics positivity monotonicity convexity of data among the properties that the spline for curves and surfaces need to satisfy smoothness and shape preservation of given data are mostly needed by all the designers in this paper a rational cubic function with three shape parameters has been developed data dependent sufficient constraints are derived for one of these shape parameters to preserve the inherited shape feature like monotonicity of data remaining two shape parameters are left free for designer to refine the shape of the monotone curve as desired numerical examples and interpolation error analysis show that the interpolant is not only c 2 local computationally economical and visually pleasant but also smooth the error of rational cubic function is also calculated when the arbitrary function being interpolated is c 3 in an interpolating interval the order of approximation of interpolant is o h i 3

skew symmetric nitsche s formulation in isogeometric analysis dirichlet and symmetry conditions patch coupling and frictionless contact abstract a simple skew symmetric nitsche s formulation is introduced into the framework of isogeometric analysis iga to deal with various problems in small strain elasticity essential boundary conditions symmetry conditions for kirchhoff plates patch coupling in statics and in modal analysis as well as signorini contact conditions for linear boundary or interface conditions the skew symmetric formulation is parameter free for contact conditions it remains stable and accurate for a wide range of the stabilization parameter several numerical tests are performed to illustrate its accuracy stability and convergence performance we investigate particularly the effects introduced by nitsche s coupling including the convergence performance and condition numbers in statics as well as the extra outlier frequencies and corresponding eigenmodes in structural dynamics we present the hertz test the block test and a 3d self contact example showing that the skew symmetric nitsche s formulation is a suitable approach to simulate contact problems in iga

domain decomposition for linear elasticity with dg jumps and mortars in this article we present a domain decomposition method for solving a linear elasticity system data are transmitted by jumps as in a discontinuous galerkin method and mortars are introduced at the interfaces to dissociate the computation between neighboring subdomains a decoupling algorithm condenses the unknowns on the interface the matrix of the system is constructed by parallel computation numerical experiments confirming the theory are presented

error estimates for the three field formulation with bubble stabilization in this paper we prove convergence and error estimates for the so called 3 field formulation using piecewise linear finite elements stabilized with boundary bubbles optimal error bounds are proved in l 2 and in the broken h 1 norm for the internal variable u and in suitable weighted l 2 norms for the other variables a and

a preconditioner for the feti dp formulation with mortar methods in two dimensions in this paper we consider a dual primal feti feti dp method for elliptic problems on nonmatching grids the feti dp method is a domain decomposition method that uses lagrange multipliers to match solutions continuously across subdomain boundaries in the sense of dual primal variables we use the mortar matching condition as the continuity constraints for the feti dp formulation we construct a preconditioner for the feti dp operator and show that the condition number of the preconditioned feti dp operator is bounded by c max i 1 ldots n left 1 log left h i h i right right 2 where h i and h i are sizes of domain and mesh for each subdomain respectively and c is a constant independent of h i s and h i s we allow jumps of coefficients of elliptic problems across subdomain boundaries numerical results are included

a formulation based on localized lagrange multipliers for bem fem coupling in contact problems this paper presents a unified formulation for the combination of the finite element method fem and the boundary element method bem in 3d frictional contact problems that is based on the use of localized lagrange multipliers llms resolution methods for the contact problem between non matching meshes have traditionally been based on a direct coupling of the contacting solids using classical lagrange multipliers these methods tend to generate strongly coupled systems that require a deep knowledge of the discretization characteristics on each side of the contact zone complicating the process of mixing different numerical techniques in this work a displacement contact frame is inserted between the fe and be interface meshes discretized and finally connected to the contacting substructures using llms collocated at the mesh interface nodes this methodology will provide a partitioned formulation which preserves software modularity and facilitates the connection of non matching fe and be meshes

construction and application of algebraic dual polynomial representations for finite element methods on quadrilateral and hexahedral meshes abstract given a sequence of finite element spaces which form a de rham sequence we will construct dual representations of these spaces with associated differential operators which connect these spaces such that they also form a de rham sequence the dual representations also need to satisfy the de rham sequence on the domain boundary the matrix which converts primal representations to dual representations the hodge matrix is the mass or gram matrix it will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients degrees of freedom of both representations this leads to very sparse system matrices even for high order methods the differential operators for dual representations will be defined vector operations grad curl and div for primal and dual representations are both topological and do not depend on the metric i e the size and shape of the mesh or the order of the numerical method derivatives are evaluated by applying sparse incidence and inclusion matrices to the expansion coefficients of the representations as illustration of the use of dual representations the method will be applied to i a mixed formulation for the poisson problem in 3d ii it will be shown that this approach allows one to preserve the equivalence between dirichlet and neumann problems in the finite dimensional setting and iii the method will be applied to the approximation of grad div eigenvalue problem on affine and non affine meshes

a dual lagrange method for contact problems with regularized frictional contact conditions modelling micro slip abstract this paper presents an algorithm for solving quasi static non linear elasticity contact problems with friction in the context of rough surfaces here we want to model the transition from sticking to slipping also called micro slip in a physically correct way in order to reproduce measured frictional damping the popular dual mortar method is used to enforce the contact constraints in a variationally consistent way without increasing the algebraic system size the algorithm is deduced from a perturbed lagrange formulation and combined with a serial parallel iwan model this leads to a regularized saddle point problem for which a non linear complementary function and thus a semi smooth newton method can be derived numerical examples demonstrate the applicability to industrial problems and show good agreement to experimentally obtained results

unilateral non linear dynamic contact of thin walled structures using a primal dual active set strategy the efficient modelling of three dimensional contact problems is still a challenge in non linear implicit structural analysis we use a primal dual active set strategy siam j optim 2003 13 865 888 based on dual lagrange multipliers siam j numer anal 2000 38 989 1012 to handle the non linearity of the contact conditions this allows us to enforce the contact constraints in a weak integral sense without any additional parameter due to the biorthogonality condition of the basis functions the lagrange multipliers can be locally eliminated we perform a static condensation to achieve a reduced system for the displacements the lagrange multipliers representing the contact pressure can be easily recovered from the displacements in a variationally consistent way for the application to thin walled structures we adapt a three dimensional non linear shell formulation including the thickness stretch of the shell to contact problems a reparametrization of the geometric description of the shell body gives us a surface oriented shell element which allows the application of contact conditions directly to nodes lying on the contact surface shell typical locking phenomena are treated with the enhanced assumed strain method and the assumed natural strain method the discretization in time is done with the implicit generalized method j appl mech 1993 60 371 375 and the generalized energy momentum method comp methods appl mech eng 1999 178 343 366 to compare the development of energies within a frictionless contact description in order to conserve the total energy within the discretized frictionless contact framework we follow an approach from laursen and love int j numer methods eng 2002 53 245 274 who introduced a discrete contact velocity to update the velocity field in a post processing step various examples show the good performance of the primal dual active set strategy applied to the implicit dynamic analysis of thin walled structures

a new weighted rational cubic interpolation and its approximation a weighted rational cubic spline interpolation has been constructed using two kinds of rational cubic spline with quadratic denominator the degree of smoothness of this spline is c 2 in the interpolating interval when the parameters satisfy a continuous system the sufficient and necessary conditions that constrain the interpolant curves to be convex in the interpolating interval or subinterval are derived also the error estimate formulas of this interpolation are obtained

a new computational approach to contact mechanics using variable node finite elements in this paper a new computational strategy for two dimensional contact problems is developed with the aid of variable node finite elements within the range of infinitesimal deformations the variable node elements which are among mls moving least square based finite elements enable us to transform node to surface contact problems into node to node contact problems this contact formulation with variable node elements leads to an accurate and effective solution procedure needless to mention that the contact patch test is passed without any additional treatment through several numerical examples we demonstrate its simplicity and the effectiveness of the proposed scheme

a stable energy conserving approach for frictional contact problems based on quadrature formulas a common approach for the numerical simulation of non linear multi body contact problems is the use of lagrange multipliers to model the contact conditions the stability of standard algorithms is improved by introducing a modified mass matrix which assigns no mass to the potential contact nodes by this the spurious algorithmic oscillations in the multiplier do not occur any more which facilitates the application of the primal dual active set strategy to dynamical contact problems the new mass matrix is calculated via a modified quadrature formula that needs no extra computational cost in addition the conservation properties of the underlying algorithm are transferred to the modified mass version different numerical examples for frictional two body contact problems illustrate the improvement in the results for the contact stresses copyright 2007 john wiley sons ltd

a weighted nitsche stabilized method for small sliding contact on frictional surfaces abstract we propose a weighted nitsche framework for small sliding frictional contact problems on three dimensional interfaces the proposed method inherits the advantages of both augmented lagrange multiplier and penalty methods while also addressing their shortcomings algorithmic details of the traction update and consistent linearization in the presence of nitsche terms are provided several benchmark numerical experiments are conducted and the results are compared with existing studies the results are encouraging and indicate accurate satisfaction of the non interpenetration constraint stable tractions and asymptotic quadratic convergence of the newton raphson method

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

coupling discontinuous galerkin and mixed finite element discretizations using mortar finite elements discontinuous galerkin dg and mixed finite element mfe methods are two popular methods that possess local mass conservation in this paper we investigate dg dg and dg mfe domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface the subdomain grids need not match and the mortar grid may be much coarser giving a two scale method convergence results in terms of the fine subdomain scale h and the coarse mortar scale h are established for both types of couplings in addition a nonoverlapping parallel domain decomposition algorithm is developed which reduces the coupled system to an interface mortar problem the properties of the interface operator are analyzed

enabling local time stepping in the parallel implicit solution of reaction diffusion equations via space time finite elements on shallow tree meshes for many applications local time stepping offers an interesting and worthwhile alternative to the by now well established global time step control in fact local time stepping can allow for a highly detailed resolution of localized features of the solution with strongly reduced computational cost when compared to global time step control however local time stepping is not applicable in a straight forward manner in the context of fully implicit time discretizations here we present a method for the efficient parallel adaptive solution of non linear partial differential equations in particular reaction diffusion equations using spatially adapted time step sizes in the context of a fully implicit solution strategy our proposed method uses a discontinuous galerkin method in time approach within a full space time approach moreover it is designed from scratch for efficient parallel computation we employ shallow tree based mesh data structures in order to ensure a low memory footprint of the adaptive meshes by solving the time dependent partial differential equation on a d 1 dimensional non conforming mesh space time adaptivity is naturally achieved in combination with a discontinuous galerkin method in time the size of the arising systems can be precisely controlled we additionally introduce and discuss a stabilization scheme for space time mortar element methods that also has a highly positive impact on the efficiency of preconditioning techniques for the arising systems of equations we present results from extensive numerical experiments that address the question of convergence and efficiency linear and non linear solver performance parallel scalability up to 2048 cores as well as accuracy for the linear heat equation and a real world non linear reaction diffusion equation from the field of computational electrocardiology

shape preserving interpolation by parametrically defined curves criteria are discussed for interpolation which preserves local convexity and local monotonicity particular schemes are then given for interpolation by parametric piecewise polynomials the interpolating curves can have any desired order of geometric continuity and they are convex in a region where the data are convex if the data arise from a function then the curve is also single valued and in this case the curve can also preserve local monotonicity if desired

on some aspects for contact with rigid surfaces surface to rigid surface and curves to rigid surface algorithms abstract special algorithms allowing a simplified description of contact between deformable body and rigid surfaces are developed based on the geometrically exact covariant description of contact a special attention is given to various geometric combinations where the contact can be represented as a contact between surfaces and b contact between a curve and a surface for contact between surfaces leading to the segment to analytical surface stas approach two algorithms can be distinguished based on the selection of a coordinate system for the closest point projection cpp procedure a rigid surface is a slave surface and b rigid surface is a master surface a special combination of both contact kinematics for the surface to surface and for the curve to curve approaches is employed for the contact between a curve and a surface leading to the curve to rigid surface ctrs approach the last algorithm is verified with the well known euler formula for the rope cylinder interaction as well as with a new derived generalization into a 3d spiral rope on a cylinder the developed algorithms can be straightforwardly implemented within an iso geometric approach as well as within the conventional finite elements where rigid surfaces are given by cad patches any type of elements can be employed for the contacting deformable surface curve because the algorithms are formulated in a covariant form

a bivariate rational interpolation with a bi quadratic denominator in this paper a new rational interpolation with a bi quadratic denominator is developed to create a space surface using only values of the function being interpolated the interpolation function has a simple and explicit rational mathematical representation when the knots are equally spaced the interpolating function can be expressed in matrix form and this form has a symmetric property the concept of integral weights coefficients of the interpolation is given which describes the weight of the interpolation points in the local interpolating region

two scale dirichlet neumann preconditioners for elastic problems with boundary refinements abstract the present work deals with the efficient resolution of elastostatics problems on domains with boundary refinements the proposed approach separates the boundary refinements from the interior of the domain by the mortar method and uses dirichlet neumann preconditioners to solve the corresponding algebraic system we prove that the simplest dirichlet neumann algorithm achieves independence of the condition number of the preconditioned system with respect to the number and the size of the small details nevertheless the situation no longer prevails when the refined boundary is clamped an enhanced preconditioner is then designed by the introduction of a coarse space to mitigate the aforementioned sensitivity some numerical tests are performed to confirm the analysis and the tools are extended by the proposition of a quasi newton method in the case of nonlinear elasticity this paper is an extended version of a work presented at the dd16 conference with proofs and complete numerical results

a new c2 rational interpolation based on function values and constrained control of the interpolant curves in this paper a new method is developed to create a high order smoothness interpolation using values of the function being interpolated this is a kind of rational cubic interpolation with quadratic denominator this rational spline not only belongs to c 2 in the interpolating interval but could also be used to constrain the shape of the interpolant curve such as to force it to be in the given region all because of the selectable parameters in the rational spline itself the more important achievement mathematically of this method is that the uniqueness of the interpolating function for the given data would be replaced by the uniqueness of the interpolating curve for the given data and the selected parameters

a scalable feti dp algorithm with non penetration mortar conditions on contact interface by combining feti algorithms of dual primal type with recent results for bound constrained quadratic programming problems we develop an optimal algorithm for the numerical solution of coercive variational inequalities the model problem is discretized using non penetration conditions of mortar type across the potential contact interface and a feti dp algorithm is formulated the resulting quadratic programming problem with bound constraints is solved by a scalable algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem numerical experiments for non matching meshes across the contact interface confirm the theoretical scalability of the algorithm

an embedded mesh method in a multiple material ale abstract a new approach for treating the mechanical interactions of overlapping finite element meshes is presented referred to as embedded mesh methods here these overlapping mesh methods typically include a foreground solid mesh and a background euler fluid grid or solid mesh a number of different approaches have been used in previous work to characterize the interactions of the background and foreground meshes at the interface lagrange multipliers are well suited to enforce the continuity constraints but care must be taken such that the resulting formulation is stable several lagrange multiplier techniques are examined in this work and applied to coupling solid meshes and fluid structure interaction type problems in addition details regarding implementation in a two step multi material arbitrary lagrangian eulerian ale code are presented example problems demonstrate convergence and applicability to a range of problems in particular the fluid structure interaction examples focus on blast applications

a petrov galerkin formulation for the alpha interpolation of fem and fdm stencils applications to the helmholtz equation summary a new petrov galerkin pg method involving two parameters namely 1 and 2 is presented which yields the following schemes on rectangular meshes i a compact stencil obtained by the linear interpolation of the galerkin fem and the classical central finite difference method fdm should the parameters be equal that is 1 2 and ii the nonstandard compact stencil presented in int j numer meth engng 2011 86 18 46 for the helmholtz equation if the parameters are distinct that is 1 2 the nonstandard compact stencil is obtained by taking the linear interpolation of the diffusive terms specified by 1 and the mass terms specified by 2 that appear in the stencils obtained by the standard galerkin fem and the classical central fdm respectively on square meshes these two schemes were shown to provide solutions to the helmholtz equation that have a dispersion accuracy of fourth and sixth order respectively int j numer meth engng 2011 86 18 46 the objective of this paper is to study the performance of this pg method for the helmholtz equation using nonuniform meshes and the treatment of natural boundary conditions copyright 2011 john wiley sons ltd

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

a time discretization scheme based on rothe s method for dynamical contact problems with friction abstract we present a new dissipative and contact stabilized time discretization scheme for dynamic frictional contact problems which is based on rothe s method especially for the case of coulomb friction the stability of the contact stresses is of crucial importance as they directly influence the frictional behavior in our approach we obtain an accurate description of the frictional stresses by a time discretized friction law allowing for an implicit treatment of the contact forces in the framework of the newmark scheme moreover undesirable oscillations at the contact interface are removed by employing an additional l 2 projection within the predictor step since the implicit treatment of the material behavior and the frictional response requires a quasi variational inequality to be solved in each time step we derive a non smooth multiscale method which allows for the efficient and robust solution of these highly non linear problems the convergence of this multiscale method is proven for the case of tresca friction for the case of coulomb friction an inexact fixed point iteration in the normal stresses is used we furthermore consider the case of two body contact here the information transfer at the contact interface is realized by means of mortar methods which provide a stable discretization of the relative displacements and the stresses at the contact boundary numerical results for the resulting fully discrete scheme in 3d are presented showing the high accuracy of the proposed method

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

information transfer between incompatible finite element meshes application to coupled thermo viscoelasticity this article is concerned with information transfer between non matching finite element meshes such a feature is not new in the literature but we focus herein on a geometric approach to transfer solution fields in order to be as application independent as possible moreover the dedicated case concerns the transfer of finite element fields defined at integration points of the meshes and allows iterative exchange of fields in both directions to do so we propose an extension to the mortar technique that fulfills these goals and that does not suffer from a high computational cost the application is the simulation of a strongly coupled thermo viscoelastic problem with phase transition solved with a partitioning technique

explicit synchronous partitioned algorithms for interface problems based on lagrange multipliers abstract traditional explicit partitioned schemes exchange boundary conditions between subdomains and can be related to iterative solution methods for the coupled problem as a result these schemes may require multiple subdomain solves acceleration techniques or optimized transmission conditions to achieve sufficient accuracy and or stability we present a new synchronous partitioned method derived from a well posed mixed finite element formulation of the coupled problem we transform the resulting differential algebraic equation dae to a hessenberg index 1 form in which the algebraic equation defines the lagrange multiplier as an implicit function of the states using this fact we eliminate the multiplier and reduce the dae to a system of explicit odes for the states explicit time integration both discretizes this system in time and decouples its equations as a result the temporal accuracy and stability of our formulation are governed solely by the accuracy and stability of the explicit scheme employed and are not subject to additional stability considerations as in traditional partitioned schemes we establish sufficient conditions for the formulation to be well posed and prove that classical mortar finite elements on the interface are a stable choice for the lagrange multiplier we show that in this case the condition number of the schur complement involved in the elimination of the multiplier is bounded by a constant the paper concludes with numerical examples illustrating the approach for two different interface problems

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

iterative substructuring preconditioners for mortar element methods in two dimensions the mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains the resulting methods are nonconforming but still yield optimal approximations in this paper we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric second order elliptic equations in two dimensions both spectral and finite element methods for geometrically conforming as well as nonconforming domain decompositions are studied in each case we obtain a polylogarithmic bound on the condition number of the preconditioned matrix

domain decomposition with nonmatching grids augmented lagrangian approach we propose and study a domain decomposition method which treats the constraint of displacement continuity at the interfaces by augmented la grangian techniques and solves the resulting problem by a parallel version of the peaceman rachford algorithm we prove that this algorithm is equivalent to the fictitious overlapping method introduced by p l lions we also prove its linear convergence independently of the discretization step h even if the finite element grids do not match at the interfaces a new preconditioner using fictitious overlapping and well adapted to three dimensional elasticity problems is also introduced and is validated on several numerical examples

an augmented lagrangian technique combined with a mortar algorithm for modelling mechanical contact problems summary a finite element formulation for three dimensional 3d contact mechanics using a mortar algorithm combined with a mixed penalty duality formulation from an augmented lagrangian approach is presented in this method no penalty parameter is introduced for the regularisation of the contact problem the contact approach based on the mortar method gives a smooth representation of the contact forces across the bodies interface and can be used in arbitrarily curved 3d configurations the projection surface used for integrating the equations is built using a local cartesian basis defined in each contact element a unit normal to the contact surface is defined locally at each element simplifying the implementation and linearisation of the equations the displayed examples show that the algorithm verifies the contact patch tests exactly and is applicable to large displacements problems with large sliding motions copyright 2012 john wiley sons ltd

analysis of a space time discretization for dynamic elasticity problems based on mass free surface elements in this paper a new space time discretization is proposed which is based on a modified mass matrix the mass associated with a surface layer of elements is redistributed such that the inertia at the boundary is removed this approach is motivated by the observation that standard space time discretization schemes applied to dynamic contact problems yield spurious oscillations a widely used approach for the numerical simulation of these problems is based on lagrange multipliers which represent the contact stresses but the algebraic contact conditions in combination with the inertia volume terms often yield nonphysical results for the contact stresses and the stability of the algorithm can be lost our modified matrix is calculated via nonstandard quadrature formulas that require no extra computational effort in addition the conservation properties of the underlying algorithm are carried over to the modified method and the standard optimal a priori estimates are still satisfied numerical examples confirm the optimality of the approach and its stabilization effect applied to contact problems

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

a weighted bivariate blending rational interpolation based on function values abstract this paper presents a new weighted bivariate blending rational spline interpolation based on function values this spline interpolation has the following advantages firstly it can modify the shape of the interpolating surface by changing the parameters under the condition that the values of the interpolating nodes are fixed secondly the interpolating function is c 1 continuous for any positive parameters thirdly the interpolating function has a simple and explicit mathematical representation fourthly the interpolating function only depends on the values of the function being interpolated so the computation is simple in addition this paper discusses some properties of the interpolating function such as the bases of the interpolating function the matrix representation the bounded property the error between the interpolating function and the function being interpolated