the arlequin method as a flexible engineering design tool by superposing and gluing models the arlequin method offers an extended modelling framework for the design of engineering structures this paper aims at developing the numerical aspects of the approach and at showing how it can be used with great flexibility and in a consistent manner to change locally a global mechanical model the capabilities of the arlequin method and the effectiveness of the implemented numerical tools are exemplified by 1 d 2 d and 3 d numerical applications

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

a subspace of the dssy nonconforming quadrilateral finite element space for the stokes equations in this paper we propose a subspace of the dssy nonconforming quadrilateral finite element space the product of this space together with the piecewise constant space can be used for approximating the velocity and pressure variables respectively in solving stokes problems more precisely this space consists of the p 1 nonconforming quadrilateral finite element space augmented by macro bubble functions based on the dssy nonconforming quadrilateral space under a hood taylor type assumption on meshes it is shown that the pair satisfies the discrete inf sup condition using a boundedness estimate of an interpolation operator based on edge integrals numerical results are presented

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

a conforming finite element method for overlapping and nonmatching grids in this paper we propose a finite element method for nonmatching overlapping grids based on the partition of unity both overlapping and nonoverlapping cases are considered we prove that the new method admits an optimal convergence rate the error bounds are in terms of local mesh sizes and they depend on neither the overlapping size of the subdomains nor the ratio of the mesh sizes from different subdomains our results are valid for multiple subdomains and any spatial dimensions

on the modelling of incompressibility in linear and non linear elasticity with the master slave approach this is the pre peer reviewed version of the following article munoz jose j on the modelling of incompressibility in linear and non linear elasticity with the master slave approach international journal for numerical methods in engineering abril 2008 vol 74 num 2 p 269 293 which has been published in final form at http www3 interscience wiley com journal 116323442 abstract

parallel spectral element solution of the stokes problem abstract in this paper we present a high efficiency medium grained parallel spectral element method for numerical solution of the stokes problem in general domains the method is based upon naturally concurrent uzawa and jacobi preconditioned conjugate gradient iterative methods geometry based data parallel distribution of work amongst processors nearest neighbor sparsity and high order substructuring for minimum communication general locally structured globally unstructured parallel constructs and efficient embedding of vector reduction operations for inner product and norm calculation an analysis is given for the computational complexity of the algorithm on a native medium grained parallel processor and the potential communication superiority of high order discretizations is described lastly the method is implemented on the fast intel vector hypercube and the performance of this algorithm architecture coupling is evaluated in a technical and economic framework that reflects the true advantages of parallel solution of partial differential equations

a new paradigm for parallel adaptive meshing algorithms we present a new approach to the use of parallel computers with adaptive finite element methods this approach addresses the load balancing problem in a new way requiring far less communication than current approaches it also allows existing sequential adaptive pde codes such as pltmg and mc to run in a parallel environment without a large investment in recoding in this new approach the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor using a sequential adaptive solver without requiring any modifications to the sequential solver the small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh which are then used in a weighted recursive spectral bisection of the initial mesh the bulk of the calculation then takes place independently on each processor with no communication using possibly the same sequential adaptive solver each processor adapts its region of the mesh independently and a nearly load balanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection only the initial fan out of the mesh decomposition to the processors requires communication two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution namely the construction of a global conforming mesh from the independent subproblems followed by a final smoothing phase using the subdomain solutions as an initial guess we present a series of convincing numerical experiments which illustrate the effectiveness of this approach the justification of the initial refinement prediction step as well as the justification of skipping the two communication intensive steps is supported by some recent j xu and a zhou math comp to appear and not so recent j a nitsche and a h schatz math comp 28 1974 pp 937 958 a h schatz and l b wahlbin math comp 31 1977 pp 414 442 a h schatz and l b wahlbin math comp 64 1995 pp 907 928 results on local a priori and a posteriori error estimation

superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data a semidiscrete mixed finite element approximation to parabolic initial boundary value problems is introduced and analyzed superconvergence estimates for both pressure and velocity are obtained the estimates for the errors in pressure and velocity depend on the smoothness of the initial data including the limiting cases of data in l 2 and data in h r for r sufficiently large because of the smoothing properties of the parabolic operator these estimates for large time levels essentially coincide with the estimates obtained earlier for smooth solutions however for small time intervals we obtain the correct convergence orders for nonsmooth data

stabilized low order finite elements for failure and localization problems in undrained soils and foundations abstract geomaterials in general and soils in particular are highly nonlinear materials presenting a very strong coupling between solid skeleton and intersticial water in the limit of zero compressibility of water and soil grains and zero permeability which correspond to the classical undrained assumption of soil mechanics the functions used to interpolate displacements and pressures must fulfill either the babuska brezzi conditions or the much simpler patch test proposed by zienkiewicz and taylor these requirements exclude the use of elements with equal order interpolation for pressures and displacements for which spurious oscillations may appear the simplest elements with continuous pressures which can be used in 2d are the quadratic triangle and quadrilateral with linear and bilinear pressures respectively the purpose of this paper is to present a stabilization technique allowing the use of both linear triangles for displacements and pressures t3p3 and bilinear quadrilaterals q4p4 the proposed element will be applied to obtain limit loads and failure surfaces in simple boundary value problems for which analytical solutions exist

primal dual weak galerkin finite element methods for elliptic cauchy problems abstract the authors propose and analyze a well posed numerical scheme for a type of ill posed elliptic cauchy problem by using a constrained minimization approach combined with the weak galerkin finite element method the resulting euler lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable and is thus an example of the primal dual weak galerkin finite element method this new primal dual weak galerkin algorithm is consistent in the sense that the system is symmetric well posed and is satisfied by the exact solution a certain stability and error estimates were derived in discrete sobolev norms including one in a weak l 2 topology some numerical results are reported to illustrate and validate the theory developed in the paper

the mortar element method with overlapping subdomains in this paper we discuss an extension of the mortar element method to overlapping subdomains and nonmatching grids in the overlapped zones we discuss in particular the case where more than two subdomains overlap the method is described and analyzed as are some preconditioned iterative methods for solving the linear systems arising from this discretization

west s problem on equivariant hyperspaces and banach mazur compacta let g be a compact lie group x a metric g space and exp x the hyperspace of all nonempty compact subsets of x endowed with the hausdorff metric topology and with the induced action of g we prove that the following three assertions are equivalent a x is locally continuum connected resp connected and locally continuum connected b expx is a g anr resp a g ar c exp x g is an anr resp an ar this is applied to show that expg g is an anr resp an ar for each compact resp connected lie group g if g is a finite group then expx g is a hilbert cube whenever x is a nondegenerate peano continuum let l n be the hyperspace of all centrally symmetric compact convex bodies a c r n n 2 for which the ordinary euclidean unit ball is the ellipsoid of minimal volume containing a and let l 0 n be the complement of the unique o n fixed point in l n we prove that 1 for each closed subgroup h c o n l 0 n h is a hilbert cube manifold 2 for each closed subgroup k c o n acting non transitively on s n 1 the k orbit space l n k and the k fixed point set l n k are hilbert cubes as an application we establish new topological models for tha banach mazur compacta l n o n and prove that l 0 n and exp s n 1 s n 1 have the same o n homotopy type

coupled fe be method for eigenvalue analysis of elastic structures submerged in an infinite fluid domain summary for thin elastic structures submerged in heavy fluid e g water a strong interaction between the structural domain and the fluid domain occurs and significantly alters the eigenfrequencies therefore the eigenanalysis of the fluid structure interaction system is necessary in this paper a coupled finite element and boundary element fe be method is developed for the numerical eigenanalysis of the fluid structure interaction problems the structure is modeled by the finite element method the compressibility of the fluid is taken into consideration and hence the helmholtz equation is employed as the governing equation and solved by the boundary element method bem the resulting nonlinear eigenvalue problem is converted into a small linear one by applying a contour integral method adequate modifications are suggested to improve the efficiency of the contour integral method and avoid missing the eigenvalues of interest the burton miller formulation is applied to tackle the fictitious eigenfrequency problem of the bem and the optimal choice of its coupling parameter is investigated for the coupled fe be method numerical examples are given and discussed to demonstrate the effectiveness and accuracy of the developed fe be method copyright 2016 john wiley sons ltd

fast bem fem mortar coupling for acoustic structure interaction a coupling algorithm based on lagrange multipliers is proposed for the simulation of structure acoustic field interaction finite plate elements are coupled to a galerkin boundary element formulation of the acoustic domain the interface pressure is interpolated as a lagrange multiplier thus allowing the coupling of non matching grids the resulting saddle point problem is solved by an approximate uzawa type scheme in which the matrix vector products of the boundary element operators are evaluated efficiently by the fast multipole boundary element method the algorithm is demonstrated on the example of a cavity backed elastic panel

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

conservative load transfer along curved fluid solid interface with non matching meshes we investigate the effect of curvature on the accuracy of schemes used to transfer loads along the interface in coupled fluid solid simulations involving non matching meshes we analyze two types of load transfer schemes for the coupled system a point to element projection schemes and b common refinement schemes the accuracy of these schemes over the curved interface is assessed with the aid of static and transient problems we show that the point to element projection schemes may yield inaccurate load transfer from the source fluid mesh to the target solid mesh leading to a weak instability in the form of spurious oscillations and overshoots in the interface solution the common refinement scheme resolves this problem by providing an accurate transfer of discrete interface conditions across non matching meshes we show theoretically that the accurate transfer preserves the stability of the coupled system while maintaining the energy conservation over a reference interface finally we introduce simple analytical error functions which correlate well with the numerical errors of the load transfer schemes

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

overlapping nonmatching grid mortar element methods for elliptic problems in the first part of the paper we introduce an overlapping mortar finite element method for solving two dimensional elliptic problems discretized on overlapping nonmatching grids we prove an optimal error bound and estimate the condition numbers of certain overlapping schwarz preconditioned systems for the two subdomain case we show that the error bound is independent of the size of the overlap and the ratio of the mesh parameters in the second part we introduce three additive schwarz preconditioned conjugate gradient algorithms based on the trivial and harmonic extensions we provide estimates for the spectral bounds on the condition numbers of the preconditioned operators we show that although the error bound is independent of the size of the overlap the condition number does depend on it numerical examples are presented to support our theory

inf sup conditions for the mortar spectral element discretization of the stokes problem summary the aim of this paper is to prove some babuska brezzi type conditions which are involved in the mortar spectral element discretization of the stokes problem for several cases of nonconforming domain decompositions

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

a new domain decomposition method with overlapping patches for ultrascale simulations application to biological flows we address the failure in scalability of large scale parallel simulations that are based on semi implicit time stepping and hence on the solution of linear systems on thousands of processors we develop a general algorithmic framework based on domain decomposition that removes the scalability limitations and leads to optimal allocation of available computational resources it is a non intrusive approach as it does not require modification of existing codes specifically we present here a two stage domain decomposition method for the navier stokes equations that combines features of discontinuous and continuous galerkin formulations at the first stage the domain is subdivided into overlapping patches and within each patch a c 0 spectral element discretization second stage is employed solution within each patch is obtained separately by applying an efficient parallel solver proper inter patch boundary conditions are developed to provide solution continuity while a multilevel communicating interface mci is developed to provide efficient communication between the non overlapping groups of processors of each patch the overall strong scaling of the method depends on the number of patches and on the scalability of the standard solver within each patch this dual path to scalability provides great flexibility in balancing accuracy with parallel efficiency the accuracy of the method has been evaluated in solutions of steady and unsteady 3d flow problems including blood flow in the human intracranial arterial tree benchmarks on bluegene p cray xt5 and sun constellation linux cluster have demonstrated good performance on up to 96 000 cores solving up to 8 21b degrees of freedom in unsteady flow problem the proposed method is general and can be potentially used with other discretization methods or in other applications

a stabilized explicit lagrange multiplier based domain decomposition method for parabolic problems a fully explicit stabilized domain decomposition method for solving moderately stiff parabolic partial differential equations pdes is presented writing the semi discretized equations as a differential algebraic equation dae system where the interface continuity constraints between subdomains are enforced by lagrange multipliers the method uses the runge kutta chebyshev projection scheme to integrate the dae explicitly and to enforce the constraints by a projection with mass lumping techniques and node to node matching grids the method is fully explicit without solving any linear system a stability analysis is presented to show the extended stability property of the method the method is straightforward to implement and to parallelize numerical results demonstrate that it has excellent performance

the schwarz alternating method in solid mechanics abstract we advance the schwarz alternating method as a means for concurrent multiscale coupling in finite deformation solid mechanics we prove that the schwarz alternating method converges to the solution of the problem on the entire domain and that the convergence rate is geometric provided that each of the subdomain problems is well posed i e their corresponding energy density functions are quasi convex it is shown that the use of a newton type method for the solution of the resultant nonlinear system leads to two kinds of block linearized systems depending on the treatment of the dirichlet boundary conditions the first kind is a symmetric block diagonal linear system in which each diagonal block is the tangent stiffness of each subdomain i e the off diagonal blocks are all zero and the coupling terms appear only on the right hand side the second kind is a nonsymmetric block system with off diagonal coupling terms several variants of the schwarz alternating method are proposed for the first kind of linear system including one in which the schwarz alternating iterations and the newton iterations are combined into a single scheme this version of the method is particularly attractive as it lends itself to a minimally intrusive implementation into existing finite element codes finally we demonstrate the performance of the proposed variants of the schwarz alternating method on several one dimensional and three dimensional examples

stabilized discontinuous finite element approximations for stokes equations in this paper we derive two stabilized discontinuous finite element formulations symmetric and nonsymmetric for the stokes equations and the equations of the linear elasticity for almost incompressible materials these methods are derived via stabilization of a saddle point system where the continuity of the normal and tangential components of the velocity displacements are imposed in a weak sense via lagrange multipliers for both methods almost all reasonable pair of discontinuous finite element spaces can be used to approximate the velocity and the pressure optimal error estimate for the approximation of both the velocity of the symmetric formulation and pressure in l2 norm are obtained as well as one in a mesh dependent norm for the velocity in both symmetric and nonsymmetric formulations

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

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

wavelet stabilization of the lagrange multiplier method we propose here a stabilization strategy for the lagrange multiplier formulation of dirichlet problems the stabilization is based on the use of equivalent scalar products for the spaces h 1 2 partial omega and h 1 2 partial omega which are realized by means of wavelet functions the resulting stabilized bilinear form is coercive with respect to the natural norm associated to the problem a uniformly coercive approximation of the stabilized bilinear form is constructed for a wide class of approximation spaces for which an optimal error estimate is provided finally a formulation is presented which is obtained by eliminating the multiplier by static condensation this formulation is closely related to the nitsche s method for solving dirichlet boundary value problems

mortar based frictional contact formulation for higher order interpolations using the moving friction cone abstract finite element discretizations including contact usually apply the standard nts node to segment element thus a coupling with higher order solid elements leads to inconsistencies in the transmission of contact stresses one can circumvent this using the mortar method which includes a weak projection of the contact constraints in this paper we present a penalty formulation based on the mortar concept for two dimensional large deformation frictional contact the discretization of the contact surfaces contains quadratic approximations which can be used within a quadratic approximation of the solid elements to obtain a simple and efficient matrix formulation the moving friction cone algorithm is applied

hyperspaces of smooth convex bodies up to congruence abstract we determine the homeomorphism type of the hyperspace of positively curved c convex bodies in r n and derive various properties of its quotient by the group of euclidean isometries we make a systematic study of hyperspaces of convex bodies that are at least c 1 we show how to destroy the symmetry of a family of convex bodies and prove that this cannot be done modulo congruence

flow rate boundary problems for an incompressible fluid in deformable domains formulations and solution methods in this paper we consider the numerical solution of the interaction of an incompressible fluid and an elastic structure in a truncated computational domain as well known in this case there is the problem of prescribing realistic boundary data on the artificial sections when only partial data are available this problem has been investigated extensively for the rigid case in this work we start considering the compliant case by focusing on the flow rate conditions for the fluid we propose three formulations of this problem different algorithms for its numerical solution and carry out several 2d numerical simulations with the aim of comparing the performances of the different algorithms

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

hybrid finite element methods for the signorini problem we study three mixed linear finite element methods for the numerical simulation of the two dimensional signorini problem applying falk s lemma and saddle point theory to the resulting discrete mixed variational inequality allows us to state the convergence rate of each of them two of these finite elements provide optimal results under reasonable regularity assumptions on the signorini solution and the numerical investigation shows that the third method also provides optimal accuracy

on the symmetric square unstable local transfer we prove the fundamental lemma for spherical functions with respect to the natural induction lifting frompgl 2 topgl 3 which appears as the unstable counterpart of the stable symmetric square lifting fromsl 2 topgl 3 see iv for an introduction to this project and vi for the final results thus spherical functions onpgl 2 andpgl 3 which correspond to each other by satisfying an elementary representation theoretic relation are shown to have matching orbital integrals the proof of this local statement is based on an application of the global trace formula

stability and convergence of nonconforming hp finite element methods the stability and convergence of nonconforming hp finite element methods in par ticular the mortar finite element method and its variants are established based on a new stability measure for these methods using a generalized eigenvalue analysis estimates for this measure are computed numerically our numerical results demonstrate that these nonconforming methods prove to be good candidates for hp implementation and also behave ss well as conforming finite element methods the discussion here ls primarily in two dimensions but some extensions to three dimensions

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

maximum norm stability of difference schemes for parabolic equations on overset nonmatching space time grids in this paper theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space time grids we consider parabolic equations containing a linear reaction term on a space time domain 0 t which is decomposed into an overlapping collection of cylindrical subregions of the form l 0 t for l 1 p each of the space time domains l 0 t are assumed to be independently grided in parallel according to the local geometry and space time regularity of the solution yielding space time grids with mesh parameters hl and l in particular the different space time grids need not match on the regions of overlap and the time steps l can differ from one grid to the next we discretize the parabolic equation on each local grid by employing an explicit or implicit scheme in time and a finite difference scheme in space satisfying a discrete maximum principle the local discretizations are coupled together without the use of lagrange multipliers by requiring the boundary values on each space time grid to match a suitable interpolation of the solution on adjacent grids the resulting global discretization yields a large system of coupled equations which can be solved by a parallel schwarz iterative procedure requiring some communication between adjacent subregions our analysis employs a contraction mapping argument applications of the results are briefly indicated for reaction diffusion equations with contractive terms and heterogeneous hyperbolic parabolic approximations of parabolic equations

a matrix free chimera approach based on dirichlet dirichlet coupling for domain composition purposes abstract a novel approach using an algebraic based chimera type domain composition method in the context of the finite element method for non matching overlapping unstructured grids is proposed in this work the scheme is based on both the transfer of information across each grid interface via dirichlet boundary conditions and a high order interpolation algorithm to obtain one global solution of the system the solution can be obtained iteratively with a convergence rate that is similar to that obtained with an analogous conformal mesh and the matrix vector operator can be computed with completely decoupled operations on both meshes furthermore the scheme can be set as a linear operator that can be fed to a matrix free efficient iterative solver such as the biconjugate gradient stabilized method several numerical examples using non matching unstructured grids that are partially and completely overlapped with different element sizes are presented assessing the precision and convergence rate of the method

on the symmetric square twisted trace formula abstract we compute a trace formula for a test function on pgl 3 a twisted by the outer automorphism g j t g 1 j j 0 1 1 1 0 the resulting formula is then compared with trace formulae for h h 0 0 sl 2 and h 1 pgl 2 and matching functions f 0 and f 1 thereof we obtain a trace formula identity which plays a key role in our study of the symmetric square lifting from h a to g a our formulae are remarkably simple due to the introduction of a new concept of a regular function this eliminates the singular and weighted integrals in the trace formulae

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

corrigendum to west s problem on equivariant hyperspaces and banach mazur compacta in our article 1 on p 3389 the definition of the weak topology of the g nerve n ja contains a gap namely it is claimed there that the topology on n ja induced from j is the weak one which is false the author apologizes for this mistake nevertheless in the proofs of lemmas 4 2 4 4 and 5 2 where the topology of m ja is essential in fact the right weak topology of m ja was applied thus all of the proofs given in 1 are correct and complete using the notation and references adopted in 1 the above mentioned gap in the definition of the topology of the g nerve m ja may be filled by replacing the text on p 3389 starting in line 26 and ending in line 35 by the following for every simplex l o un c m ja set

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

on the symmetric square orbital integrals j tg 1j for g in g the elements fi 8 of g f are called stably a conjugate if there is h in g f resp g with h a h1 conjugacy and stable conjugacy is defined analogously for h f and iii f on omitting a in sect 1 we show that the map n described in 1 5 below is a well defined bijection called the norm map from the set of stable a conjugacy classes in g f to the set of stable conjugacy classes in it f we also define a surjection n1 from the subset of this set described in 1 2 3 4 to the set of stable conjugacy classes in hi f in the case 4 n1 actually relates conjugacy classes let be an element of g f the set of eigenvalues of a is of the form 2 1 2 1 1 if 6a 1 put n 5 1 and ni 1 2 if 6a 6 is a non trivial unipotent let n6 and ni be the non trivial unipotent classes 3 if the eigenvalues 2 1 2 1 of 6a 6 are distinct let n6 be the class in h f determined by the eigenvalues 2 2 1 and ni6 the class in hi f with eigenvalues 4 1

coupling finite element and spectral methods first results a poisson equation on a rectangular domain is solved by coupling two methods the domain is divided in two squares a finite element approximation is used on the first square and a spectral discretization is used on the second one two kinds of matching conditions on the interface are presented and compared in both cases error estimates are proved

superconvergence of least squares methods for a coupled system of elliptic equations abstract we present a numerical method for solving a coupled system of elliptic partial differential equations pdes our method is based on the least squares ls approach we develop ellipticity estimates and error bounds for the method the main idea of the error estimates is the establishment of supercloseness of the ls solutions and solutions of the mixed finite element methods and ritz projections using the supercloseness property we obtain l 2 norm error estimates and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces moreover we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions application to optimal control problems is briefly considered

an iterative stabilized fractional step algorithm for finite element analysis in saturated soil dynamics abstract the discretization of the u p model for saturated porous media results in the semi discrete system of mixed type in displacements and pressures the babuska brezzi condition or the simpler patch test proposed by zienkiewicz and taylor precludes the use of elements with the equal low order of interpolation for u and p unless special stabilization techniques such as the fractional step algorithms are used the purpose of this paper is to present a modified version of the fractional step algorithm which allows much larger time step sizes than those for the existing ones the method is based on introducing an iteration algorithm the numerical experiments demonstrate the effectiveness and improved performance of the proposed modified version of the fractional step algorithm

two level finite element discretization of viscoelastic fluid flow abstract we propose and analyze a two level method of discretizing the equations of steady state flow of a viscoelastic fluid obeying an oldroyd type constitutive equation with no slip boundary condition the two level algorithm consists of solving a small non linear system of equations on the coarse mesh and then using that solution to solve a larger linear system on the fine mesh specifically following najib and sandri numer math 72 1995 223 we linearize the oldroyd type constitutive equation about the coarse mesh solution thus nullifying the difficulties brought by the advection term our theoretical error estimates show that it has optimal order accuracy provided the true solution is smooth and its norm is sufficiently small in addition our computational error estimates exhibit the validity of our analysis

on the symmetric square applications of a trace formula in this paper we prove the existence of the symmetric square lifting of admissible and of automorphic representations from the group sl 2 to the group pgl 3 complete local results are obtained relating the character of an sl 2 packet with the twisted character of self contragredient pgl 3 modules our global results relate packets of cuspidal representations of sl 2 with a square integrable component and self contragredient automorphic pgl 3 modules with a component coming from a square integrable one the sharp results which concern sl 2 rather than gl 2 are afforded by the usage of the trace formula the surjectivity and injectivity of the correspondence implies that any self contragredient automorphic pgl 3 module as above is a lift and that the space of cuspidal sl 2 modules with a square integrable component admits multiplicity one theorem and rigidity strong multiplicity one theorem for packets and not for individual representations

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

hp submeshing via non conforming finite element methods abstract non conformity in the hp version can involve incompatibility in both the degrees and the meshes between adjoining subdomains in this paper we show how the mortar finite element method m0 and two new variants m1 m2 can be used to join together such incompatible hp sub discretizations our results show optimality of the resulting non conforming method for various h p and hp discretizations including the case of exponential hp convergence over geometric meshes we also present numerical results for the lagrange multiplier when the method is implemented via a mixed method three dimensional considerations suggest that our methods m1 m2 are easier to generalize to arbitrary meshes than m0

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

on the symmetric square definitions and lemmas we define the symmetric square lifting for admissible and automorphic representations from the group h h 0 sl 2 to the group g pgl 3 and derive its basic properties this lifting is defined by means of shintani character relations the definition is suggested by the computation of orbital integrals stable and unstable in our on the symmetric square orbital integrals math ann 279 1987 173 193 it is compatible with dual group homomorphisms 0 h g and 1 h 1 g where h 1 pgl 2 the lifting is proven for induced trivial and special representations and both spherical functions and orthogonality relations of characters are studied

maximum norm analysis of overlapping nonmatching grid discretizations of elliptic equations in this paper we provide a maximum norm analysis of a finite difference scheme defined on overlapping nonmatching grids for second order elliptic equations we consider a domain which is the union of p overlapping subdomains where each subdomain has its own independently generated grid the grid points on the subdomain boundaries need not match the grid points from adjacent subdomains to obtain a global finite difference discretization of the elliptic problem we employ standard stable finite difference discretizations within each of the overlapping subdomains and the different subproblems are coupled by enforcing continuity of the solutions across the boundary of each subdomain by interpolating the discrete solution on adjacent subdomains if the subdomain finite difference schemes satisfy a strong discrete maximum principle and if the overlap is sufficiently large we show that the global discretization converges in optimal order corresponding to the largest truncation errors of the local interpolation maps and discretizations our discretization scheme and the corresponding theory allows any combination of lower order and higher order finite difference schemes in different subdomains we describe also how the resulting linear system can be solved iteratively by a parallel schwarz alternating method or a schwarz preconditioned krylov subspace iterative method several numerical results are included to support the theory

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

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

a gradient method approach to optimization based multidisciplinary simulations and nonoverlapping domain decomposition algorithms it has been shown recently that optimization based nonoverlapping domain decomposition algorithms are connected to many well known algorithms using a gradient type iterative strategy for the optimization problem we present further discussion on how to develop various algorithms that can integrate subdomain solvers into a solver for the problem in the whole domain in particular the algorithms we discuss can be used to develop efficient solvers of multidisciplinary problems which are constructed using existing subdomain solvers without the need for making changes in the latter

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

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

multilevel schwarz methods for elliptic problems with discontinuous coefficients in three dimensions multilevel schwarz methods are developed for a conforming finite element approximation of second order elliptic problems we focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions we establish a condition number estimate for the iterative operator which is independent of the coefficients and grows at most as the square of the number of levels we also characterize a class of distributions of the coefficients called quasi monotone for which the weighted l 2 projection is stable and for which we can use the standard piecewise linear functions as a coarse space in this case we obtain optimal methods i e bounds which are independent of the number of levels and subregions we also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas we consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods

multigrid methods for saddle point problems optimality systems abstract we develop multigrid methods for an elliptic distributed optimal control problem on convex domains that are robust with respect to a regularization parameter we prove the uniform convergence of the w cycle algorithm and demonstrate the performance of v cycle and w cycle algorithms through numerical experiments

on the accuracy of the mortar finite volume element method with lagrange multipliers abstract in this paper the mortar finite volume element mfve method with lagrange multipliers is considered for second order elliptic boundary value problems we prove l 2 error estimates which are affected by regularities in both the exact solution and the source term meanwhile the uniform convergence of the solution is obtained for the mfve method under minimal regularity assumption

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

a finite element method with overlapping meshes for free boundary axisymmetric plasma equilibria in realistic geometries existing finite element implementations for the computation of free boundary axisymmetric plasma equilibria approximate the unknown poloidal flux function by standard lowest order continuous finite elements with discontinuous gradients as a consequence the location of critical points of the poloidal flux that are of paramount importance in tokamak engineering is constrained to nodes of the mesh leading to undesired jumps in transient problems moreover recent numerical results for the self consistent coupling of equilibrium with resistive diffusion and transport suggest the necessity of higher regularity when approximating the flux map in this work we propose a mortar element method that employs two overlapping meshes one mesh with cartesian quadrilaterals covers the vacuum chamber domain accessible by the plasma and one mesh with triangles discretizes the region outside the two meshes overlap in a narrow region this approach gives the flexibility to achieve easily and at low cost higher order regularity for the approximation of the flux function in the domain covered by the plasma while preserving accurate meshing of the geometric details outside this region the continuity of the numerical solution in the region of overlap is weakly enforced by a mortar like mapping

the gromov hausdorff hyperspace of a euclidean space abstract we investigate the hyperspace gh r n of the isometry classes of all non empty compact subsets of a euclidean space in the gromov hausdorff metric it is proved that for any n 1 gh r n is homeomorphic to the orbit space 2 r n e n of the hyperspace 2 r n of all non empty compact subsets of a euclidean space r n equipped with the hausdorff metric and the natural action of the euclidean group e n this is further applied to prove that 2 r n e n is homeomorphic to the open cone o c o n e c h b n o n where c h b n stands for the set of all a 2 r n for which the closed euclidean unit ball b n is the least circumscribed ball the chebyshev ball these results lead to determine the complete topological structure of gh r n for n 2 namely we prove that gh r n is homeomorphic to the hilbert cube with a removed point we also prove that for n 2 gh b n is homeomorphic to the hilbert cube

on micro to macro connections in domain decomposition multiscale methods abstract micro to macro connection techniques constitute a key ingredient in the formulation of multiscale strategies in this contribution several connection methods are explored for the concurrent multiscale analysis of brittle heterogeneous materials particularly these techniques are investigated in a domain decomposition strong coupling multiscale framework the structural component under analysis is partitioned into a number of non overlapping domains and a fine scale resolution is assigned therein in an adaptive manner exploiting a zoom in technique mesh refinement is employed in the domains where crack coalescence and growth take place the original contribution presented in this manuscript consists in the study of different strong and weak locality constraints that connect coarse and fine resolution domains standard collocation and average compatibility are considered and serve as a basis for the development of two new interscale links the influence of different locality constraints is studied in terms of the mechanical response and the error distribution is compared to a full fine scale analysis