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

on the physical significance of higher order kinematic and static variables in a three dimensional shell formulation abstract in recent years considerable attention has been given to the development of higher order plate and shell models these models are able to approximately represent three dimensional effects while pertaining the efficiency of a two dimensional formulation due to pre integration of the structural stiffness matrix across the thickness especially the possibility to use unmodified complete three dimensional material laws within shell analysis has been a major motivation for the development of such models while the theoretical and numerical formulation of so called 7 parameter shell models including a thickness stretch of the shell has been discussed in numerous papers no thorough investigation of the physical significance of the additional kinematic and static variables coming along with the extension into three dimensions is known to the authors however realization of the mechanical meaning of these quantities is decisive for both a proper modeling of shell structures e g concerning loading and kinematic boundary conditions and a correct interpretation of the results in the present paper the significance of kinematic and static variables appearing in a 7 parameter model proposed by buchter and ramm 1992a are discussed it is shown how these quantities refine the model behavior and how they can be related to the classical variables such as curvatures and stress resultants furthermore the special role of the material law within such a formulation is addressed it is pointed out that certain requirements must hold for the variation of kinematic and static variables across the thickness to ensure correct results in this context it is found that the considered 7 parameter model can be regarded as optimal with respect to the number of degrees of freedom involved

conservation laws for conformally invariant variational problems we succeed in writing 2 dimensional conformally invariant non linear elliptic pde harmonic map equation prescribed mean curvature equations etc in divergence form these divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces from this form we can recover without the use of moving frame all the classical regularity results known for 2 dimensional conformally invariant non linear elliptic pde see hel it enables us also to establish new results in particular we solve a conjecture by e heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by s hildebrandt hil1 claiming that critical points of continuously differentiable elliptic conformally invariant lagrangian in two dimensions are continuous

a relaxation procedure for domain decomposition methods using finite elements we present the convergence analysis of a new domain decomposition technique for finite element approximations this technique was introduced in 11 and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent no global preconditioner is needed in the practice but simply single domain finite element solvers are required an optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated applications are given to both elliptic equations and incompressible stokes equations

a 4 node finite shell element for the implementation of general hyperelastic 3d elasticity at finite strains abstract in this paper a finite shell element for large deformations is presented based on extensible director kinematics the essential feature is an interface to arbitrary three dimensional material laws the non linear lagrangian formulation is based on the three field variational principle parametrized with the displacement vector enhanced green lagrangian strain tensor and second piola kirchhoff stress tensor the developed quadrilateral shell element is characterized by a course mesh accuracy and distortion insensitivity compared with bilinear displacement approaches furthermore plane stress response is approximately recovered in the asymptotic case of vanishing thickness a number of example problems investigating large deformation as well as finite strain applications are presented compressible and incompressible hyperelastic materials of the st venant kirchhoff neo hookean and mooney rivlin type are particularly used

a singularity removal theorem for yang mills fields in higher dimensions the purpose of this paper is to investigate the small energy behavior of weakly yang mills fields in rn for n 4 and in particular to extend the singularity removal theorem of uhlenbeck 7 to higher dimensions fix n 4 and let ft be some bounded domain in rn typically we shall restrict our attention to the cubes ft 1 2 n or ft 0 l n let g be a fixed finite dimensional compact lie group it will be convenient to consider g as embedded in some large unitary group u n let g be the lie algebra of g we define a connection on ft to be a section a of t ft q i e a g valued 1 form which is locally l2 for any connection a let 1 f a da aaa

domain decomposition methods for nonlinear problems in fluid dynamics abstract the numerical solution of finite element approximations of complicated two and three dimensional nonlinear problems can be a most formidable task in order to overcome this difficulty related to dimensionality domain splitting methods can be very effective particularly in view of obtaining a fast and economical conjugate gradient solver which can be used to precondition the solution of nonlinear problems by optimization methods via nonlinear least squares or weighted residual formulations a new technique of this type will be introduced and analysed and its efficiency will be discussed from numerical experiments concerning the numerical simulation of transonic flows for compressible inviscid fluids and incompressible viscous flows modelled by the navier stokes equations

extension theorems for plate elements with applications extension theorems for plate elements are established their applications to the analysis of nonoverlapping domain decomposition methods for solving the plate bending problems are presented numerical results support our theory

a hierarchic family of isogeometric shell finite elements a hierarchic family of isogeometric shell finite elements based on nurbs shape functions is presented in contrast to classical shell finite element formulations inter element continuity of at least c1 enables a unique and continuous representation of the surface normal within one nurbs patch this does not only facilitate formulation of kirchhoff love type shell models for which the standard galerkin weak form has a variational index of 2 but it also offers significant advantages for shear deformable reissner mindlin type shells and higher order shell models for a 5 parameter shell formulation with reissner mindlin kinematics a hierarchic difference vector which accounts for shear deformations is superimposed onto the rotated kirchhoff love type director of the deformed configuration this split into bending and shear deformations in the shell kinematics results in an element formulation which is free from transverse shear locking without the need to apply further remedies like reduced integration assumed natural strains or mixed finite element formulations the third member of the hierarchy is a 7 parameter model including thickness change and allowing for application of unmodified three dimensional constitutive laws the phenomenon of curvature thickness locking coming along with this kinematic extension again is automatically avoided by the hierarchic difference vector concept without any further treatment membrane locking and in plane shear locking are removed by two different approaches firstly elimination via the discrete strain gap dsg method and secondly removal of parasitic membrane strains using a hybrid mixed method based on the hellinger reissner variational principle the hierarchic kinematic structure of the three different shell formulations allows a straightforward combination of these elements within one mesh and is thus the ideal basis for a model adaptive approach

specifications of boundary conditions for reissner mindlin plate bending finite elements summary plate bending finite elements based on the reissner mindlin theory offer improved possibilities to pursue reliable finite element analyses the physical behaviour near the boundary can be modelled in a realistic manner and inherent limitations in the kirchhoff plate bending elements when modelling curved boundaries can easily be avoided however the boundary conditions used are crucial for the quality of the solution we identify the part of the reissner mindlin solution that controls the boundary layer and examine the behaviour near smooth edges and corners the presence of boundary layers of different strengths for different sets of boundary conditions is noted for a corner with soft simply supported edges the boundary layer removes the singular behaviour of kirchhoff type from the stress resultants we demonstrate the theoretical results by some numerical studies on simple plate structures which are discretized by an accurate higher order plate element the results provide guidance in choosing efficient meshes and appropriate boundary conditions in finite element analyses

an iterative procedure with interface relaxation for domain decomposition methods a domain decomposition method for second order elliptic problems is considered an iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed at each iteration a relaxation is accomplished at the subdomain interfaces in several circumstances a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found moreover when such a value is not available an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed this iterative method is then applied to the spectral collocation approximation of the differential problem the same kind of convergence results are proven many numerical experiments show the effectiveness of the method proposed here

a model of finite strain viscoplasticity based on unified constitutive equations theoretical and computational considerations with applications to shells abstract a model of multiplicative finite strain viscoplasticity is developed hereby the elastic internal potential is assumed to depend quadratically on the elastic strain measure ce by means of corresponding invariants hence possible extension of the model to take anisotropy into account is straightforward the evolution equation of the unified type due to bodner and partom asme j appl mech 42 1975 385 are modified so as to fit into the framework adopted the numerical treatment of the problem is fully developed specifically the algorithmic aspects of the resulting non additive structure are intensively discussed hereby the treatment of the exponential map of a non symmetric argument is of a central importance two consistent methods are developed various applications to shell problems are considered hereby a shell theory with seven degrees of freedom together with a 4 node enhanced strain finite element formulation are used a central feature of the shell formulation is its eligibility to the application of a three dimensional constitutive law

a shell theory with scale effects and higher order gradients abstract this works follows a generalised continuum framework developed by sansour 1998 to derive a strain gradient formulation suitable to address scale effects of structures where one dimension is very small e g thin films nano tubes etc whereas a previous strain gradient approach by sansour et al 2009 considered the fully three dimensional setting the approach here proposes a shell theory which aims to run computations of thin structures more efficiently and to include scale effects the theory features a generalised deformation description new strain and stress measures as consequence of these new quantities a corresponding generalised variational principle is formulated the approach is completed by dirichlet boundary conditions for the displacement field and its derivatives a numerical example is presented based on a meshfree formulation which provides the necessary c 1 continuity

energy identity for stationary yang mills given a principal bundle p rightarrow m over a riemannian manifold with compact structure group g let us consider a stationary yang mills connection a with energy int m f a 2 le lambda if we consider a sequence of such connections a i then it is understood by tian ann math 151 1 193 268 2000 that up to subsequence we can converge a i rightarrow a to a singular limit connection such that the energy measures converge f a i 2 dv g rightarrow f a 2dv g nu where nu e x d lambda n 4 is the n 4 rectifiable defect measure our main result is to show without additional assumptions that for n 4 a e point the energy density e x may be computed explicitly as the sum of the bubble energies arising from blow ups at x each of these bubbles may be realized as a yang mills connection over s 4 itself this energy quantization was proved in riviere commun anal geom 10 4 683 708 2002 assuming a uniform l 1 hessian bound on the curvatures in the sequence in fact our second main theorem is to show this hessian bound holds automatically precisely given a connection a as above we have the a priori estimate int m nabla 2 f a c lambda dim g m for the curvature it is important to note this result is proved in tandem with the energy quantization and not before it indeed we will in fact prove an effective version of the energy identity and it is this effective version which will lead to both the l 1 hessian bound and the classical energy quantization results in the course of the proof we will provide a quantitative version of the bubble tree decomposition which hold in all dimensions with effective estimates for a fixed stationary connections to produce strongest estimates in the paper we introduce an epsilon gauge condition which generalizes the usual coulomb gauge and which will exist with effective control even over singular regions on these epsilon gauges we will provide a new superconvexity estimate which will be a key tool in analyzing higher dimensional annular regions

analysis of iterative methods for the steady and unsteady stokes problem application to spectral element discretizations a new and detailed analysis of the basic uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady stokes operator is presented the paper focuses on the following new aspects explicit construction of the uzawa pressure operator spectrum for a semiperiodic model problem general relationship of the convergence rate of the uzawa procedure to classical inf sup discretization analysis and application of the method to high order variational discretization

the problem of plate modeling theoretical and computational results abstract the paper analyzes in detail the problem of various plate models properties of their solution and the question how well these models approximate the three dimensional formulation the boundary layer and corner singularities of the solution are analyzed a hierarchy of the models converging to the three dimensional solution is constructed the detailed numerical computations show various basic aspects of plate modeling in a concrete setting

a continuum based shell theory for non linear applications the paper introduces a non linear shell theory which provides a complete three dimensional state of stress since the theory is derived from simple three dimensional continuum mechanics it is very easy to understand as an example the theory is applied to quadrilateral shell elements which provide only displacement degrees of freedom located at nodes on the outer surfaces and one degree of freedom at the middle surface it is proposed to eliminate this degree of freedom on element level so that the elements have the same layout as the equivalent brick elements but have a better behaviour in bending have stress resultants and are cheaper with respect to computational effort the advantages with respect to implementation in a finite element program as well as in special applications are obvious however well known conditioning problems in thin shell applications must be expected therefore emphasis is put on this issue in the example problems it is shown that the elements can give acceptable answers in engineering applications and offer a potential for material non linear applications which will be considered in a forthcoming paper

a posteriori modeling error estimation for hierarchic plate models hierarchic plate models are dimensional reductions obtained by semidiscretization of the three dimensional plate problem in the transverse direction and energy projection we derive computable a posteriori estimators for the modeling error thus incurred under the assumption that the resulting two dimensional plate models are solved exactly the estimators are valid for homogeneous monoclinic materials for plates with unsmooth midsurfaces and for a wide class of variational edge conditions computable a posteriori bounds on the effectivity indices are also derived and sufficient conditions for the asymptotic i e as the plate thickness tends to zero and the spectral i e as the order of the plate model tends to infinity exactness of the estimators are given numerical examples are presented

homogeneous and heterogeneous domain decomposition methods for plate bending problems we consider the approximation of fourth order problems derived e g by the kirchoff plate model and the heterogeneous coupling between a fourth order problem and a reduced second order one describing e g a plate membrane model this paper is devoted to the analysis of an iteration by subdomain method the so called dirichlet neumann method which is used to solve both homogeneous and heterogeneous couplings numerical results obtained by the spectral element method are shown

on the mathematical foundation of the 1 1 2 platemodel abstract the plate problem of three dimensional 3 d linearized elastostatics is considered inthe framework of the hierarchical modelling with the help of the energy projection method it isshown that the 1 1 2 bending model is the simplest asymptotically correct model in thehierarchical family i e the distance between the solution of the 3 d problem and themodel solution in the energy norm tends to zero for vanishing plate thickness h whileusing unmodified 3 d material laws the formulation of the 1 1 2 bending model and the mechanical significance of itsingredients are discussed we present error estimates for the deviation of the 1 1 2 solutionfrom the kirchhoff solution as well as from the 3 d solution by using the 1 1 2 energy normand the 3 d energy norm respectively the analysis leads to h as order of convergence the results are illustrated by a numerical example

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

an iterative method for elliptic problems on regions partitioned into substructures some new preconditioners for discretizations of elliptic boundary problems are studied with these preconditioners the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains analytic estimates are given which guarantee that under appropriate hypotheses the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns numerical examples are presented which illustrate the theoretically predicted iterative convergence rates

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of euclidean space by ralph howard and s walter wei abstract call a compact riemannian manifold m a strongly unstable manifold if it is not the range or domain of a nonconstant stable harmonic map and also the homotopy class of any map to or from m contains elements of arbitrarily small energy if m is isometrically immersed in euclidean space then a condition on the second fundamental form of m is given which implies m is strongly unstable as compact isotropy irreducible homogeneous spaces have standard immersions into euclidean space this allows a complete list of the strongly unstable compact irreducible symmetric spaces to be made

new substructuring domain decomposition methods for advection diffusion equations abstract in this paper we consider a nonsymmetric elliptic problem and we use the techniques related to the steklov poincare operators to propose new substructuring iterative procedures in particular we propose two methods that generalize the well known neumann neumann and dirichlet neumann iterative procedures we prove that our methods that use symmetric and positive definite preconditioners lead to the construction of iterative schemes with optimal convergence properties numerical results for the finite element discretization are given

a finite element capacitance matrix method for the elliptic problem the capacitance matrix method for finite element approximation of the symmetric elliptic problem with the dirichlet condition on an arbitrary region is considered two variants of this method are presented which correspond to the situations when the single and double layer potential are used it is shown that the condition number of the capacitance matrix c is o h 1 for the first variant and the eigenvalues of c are positive and independent of h for the second variant where h is the mesh size of the triangulation

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

convergence analysis of domain decomposition algorithms with full overlapping for the advection diffusion problems the aim of this paper is to study the convergence properties of a time marching algorithm solving advection diffusion problems on two domains using incompatible discretizations the basic algorithm is first presented and theoretical or numerical results illustrate its convergence properties this study is based on spectral theory a priori estimates and a di giorgi nash maximum principle

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

a domain decomposition approach for heterogeneous time harmonic maxwell equations the time harmonic maxwell equations for a composite medium which behaves like a conductor in one part and like a perfect insulator in the other one are considered an existence and uniqueness theorem is proven for this degenerate problem in the case of dirichlet boundary conditions a finite element domain decomposition approach is then proposed for the numerical approximation of the exact solution this leads to an iteration by subdomain procedure where at each step a non degenerate boundary value problem has to be solved in each subdomain the convergence of these iterations is proven and the rate of convergence turns out to be independent of the mesh size h showing that the preconditioner implicitly defined by the iteration procedure is optimal

partial regularity for harmonic maps and related problems via gauge theory we give a new proof of partial regularity for harmonic maps in dimensions m 3 into arbitrary targets this proof avoids the use of adapted frames and permits to consider targets of minimal c 2 regularity the proof we present moreover extends to a large class of elliptic systems of quadratic growth

three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept conventional shell formulations such as 3 or 5 parameter theories or even 6 parameter theories including the thickness change as extra parameter require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness this means that the normal stress in thickness direction has to either vanish or be constant in general these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions the main objective of the present study is to introduce directly a complete 3 d constitutive law without modification therefore a 7 parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects

singular points of harmonic maps from 4 dimensional domains into 3 spheres in this article we show that if a stable stationary harmonic map u from a domain in r4 into s3 has an isolated singular point then the mapping degree deg u of u at is 1 or 1 furthermore the complete characterization of stable stationary tangent maps from b4 to s3 is given