toward a universal adaptive finite element strategy part 3 design of meshes abstract in this third paper in our series the issue of designing h p meshes which are optimal in some sense is addressed criteria for h p meshes are derived which are based on the idea of minimizing the estimated error over a mesh with a fixed number of degrees of freedom an optimization algorithm is developed based on these criteria and is applied to several model one dimensional and two dimensional elliptic boundary value problems numerical results indicate that the approach can lead to exponential rates of convergence

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

a straightforward structure to construct shape functions for variable p order meshes this paper describes a method for specifying variable p order meshes based on the topological hierarchy of mesh entities defining the finite elements a decomposition of p version mesh entity shape functions into a product of a function defined over the mesh entity and a blending function defined over the elements that the mesh entity bounds is presented the decomposed shape functions allow the effective construction of the shape functions and derivatives needed to construct element matrices for variable p enrichment

modeling of electromagnetic absorption scattering problems using hp adaptive finite elements a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for affine elements is presented the proposed method is illustrated and verified with a series of 2d experiments including elements with curved boundaries and nonhomogeneous media

finite difference methods and spatial a posteriori error estimates for solving parabolic equations in three space dimensions on grids with irregular nodes adaptive methods for solving systems of partial differential equations have become widespread much of the effort has focused on finite element methods in this paper modified finite difference approximations are obtained for grids with irregular nodes the modifications are required to ensure consistency and stability asymptotically exact a posteriori error estimates of the spatial error are presented for the finite difference method these estimates are derived from interpolation estimates and are computed using central difference approximations of second derivatives of the solution at grid nodes the interpolation error estimates are shown to converge for irregular grids while the a posteriori error estimates are shown to converge for uniform grids computational results demonstrate the convergence of the finite difference method and a posteriori error estimates for cases not covered by the theory

wamr an adaptive wavelet method for the simulation of compressible reacting flow part i accuracy and efficiency of algorithm abstract the wavelet adaptive multiresolution representation wamr method provides a robust method for controlling spatial grid adaptation fine grid spacing in regions where a solution varies greatly i e near steep gradients or near singularities and a much coarser grid where the solution varies slowly subsequently a wide range of spatial scales often demanded in challenging continuum physics problems can be efficiently captured furthermore the wavelet transform provides a direct measure of local error at each collocation point effectively producing automatically verified solutions the method is applied to the solution of unsteady compressible reactive flow equations and includes detailed diffusive transport and chemical kinetics models accuracy and performance of the method are examined on several test problems the sparse grids produced by the wamr method exhibit an impressive compression of the solution reducing the number of collocation points used by factors of many orders of magnitude when compared to uniform grids of equivalent resolution

adaptive hpq finite element methods of hierarchical models for plate and shell like structures abstract a four step adaptive strategy for hpq finite element approximations of hierarchical models of the thin plate and shell like structures is derived based on a priori and a posteriori error estimations this strategy controls three parameters the model level q the mesh size h and the approximation order p so as to produce quasi optimal hierarchical models and finite element meshes numerical results are given supporting the theoretical results

pyramid algorithms for bernstein bezier finite elements of high nonuniform order in any dimension the archetypal pyramid algorithm is the de casteljau algorithm which is a standard tool for the evaluation of bezier curves and surfaces pyramid algorithms replace an operation on a single high order polynomial by a recursive sequence of self similar affine combinations and are ubiquitous in computer aided geometric design for computations involving high order curves and surfaces pyramid algorithms have received no attention whatsoever from the high or low order finite element community we develop and analyze pyramid algorithms for the efficient handling of all of the basic finite element building blocks including the assembly of the element load vectors and element stiffness matrices the complexity of the algorithm for generating the element stiffness matrix is optimal a new nonuniform order variant of the de casteljau algorithm is developed that is applicable to the variable polynomial order case but incurs no additional complexity compared with the original algorithm the work provides the

on some hp adaptive finite element method for natural vibrations model eigenproblem results obtained by some hp adaptive fem are presented the fem is based on the texas 3 step strategy and the equilibrated residual method the focus is on numerical experiments the practical issues related to the above mentioned techniques are raised the potentials of the method for engineering applications are demonstrated

nonstationary monotone iterative methods for nonlinear partial differential equations a simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations with the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system picard gauss seidel and jacobi monotone iterative methods are presented and analyzed for the adaptive solutions the adaptive meshes are generated by the 1 irregular mesh refinement scheme which together with the m matrix of the finite element stiffness matrix lead to existence uniqueness comparison theorems with simple upper and lower solutions as initial iterates some numerical examples including a test problem with known analytical solution are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties numerical results of simulations on a mosfet with the gate length down to 34 nm are also given

adaptivity techniques for compressible inviscid flows abstract to enhance the resolution of numerically simulated flowfields grid adaptivity is vitally important further to afford complete flexibility in the spatial discretisation of the flow domain both structured unstructured and combinations of grid types are likely to be required hence the development of grid adaptivity techniques for general grids is important this paper discusses adaptivity achieved through point enrichment point derefinement and point movement applied on general grids a flexible data structure is described together with suitable strategies for adaptively refining a computational grid applications are presented of adaptive grid refinement for two dimensional compressible flow simulation

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

a fast hp adaptive finite element mesh design an optimum hp adaptive mesh design is accomplished by minimizing the number of equations for a specified error limit this new approach leads to a problem in which the h and p mesh parameters appear explicitly in the formulation the optimal conditions yields a non linear equation for each element which simultaneously supplies the optimum values for h and p parameters the methodology here developed is applied to the numerical solution of several unidimensional elliptic boundary value problems

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

on an h type mesh refinement strategy based on minimization of interpolation errors abstract an adaptive finite element method is proposed which involves an automatic mesh refinement in areas of the mesh where local errors are determined to exceed a pre assigned limit the estimation of local errors is based on interpolation error bounds and extraction formulas for highly accurate estimates of second derivatives applications to several two dimensional model problems are discussed the results indicate that the method can be very effective for both regular problems and problems with strong singularities

effective and practical h p version adaptive analysis procedures for the finite element method two practical and effective h p type finite element adaptive procedures are presented the procedures allow not only the final global energy norm error to be well estimated using hierarchic p refinement but in addition give a nearly optimal mesh the design of this is guided by the local information computed on the previous mesh the desired accuracy can always be obtained within one or at most two h p refinements the rate of convergence of the adaptive h p version analysis procedures has been tested for some examples and found to be very strong the presented procedures can easily be incorporated into existing p or h type code structures

a posteriori error estimation of steady state finite element solutions of the navier stokes equations by a subdomain residual method abstract we present a subdomain residual method as well as its mathematical basis for estimating errors in steady state finite element solutions of the incompressible navier stokes equations the estimated errors are obtained by solving a series of local problems in which velocity boundary condition is used wherever the exact traction boundary condition is not available an iterative procedure similar to the newton method is employed to improve the error estimates the performance of the method is demonstrated in two numerical examples i e the channel flow over a backward facing step and past a circular cylinder at low reynolds numbers

recent progress on application of hp adaptive be fe methods to elastic scattering summary several new issues related to the projcct described in references 1 3 are reported the presented problems include i sensitivity of p convergence results to geometric parametrizations and an idea of a new mesh generation for lip approximations 2 a posteriori error estimation for the coupled problem 3 a better understanding of the meaning of the asymptotic convergence a few typical numerical examples conclude the presentation

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985

analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes the equilibrated residual method is now accepted as the best residual type a posteriori error estimator nevertheless there remains a gap in the theory and practice of the method the present work tackles the problem of existence construction and stability of equilibrated fluxes for hp finite element approximation on hybrid meshes consisting of quadrilateral and triangular elements with hanging nodes a practical algorithm for post processing the finite element approximation is presented and shown to produce equilibrated fluxes for a general one irregular partition the resulting fluxes are shown to be stable in the sense that the associated error estimator provides a lower bound on the local error which does not degenerate with the mesh size numerical examples are included to illustrate the theoretical results

modeling of electromagnetic fields in rectangular waveguides by coupled finite infinite elements abstract the paper describes the 3d infinite element for modeling of stationary harmonic electromagnetic fields in waveguides the proposed approximation is a straightforward modification of the analogous approach developed for analysis of scattering problems in unbounded 2d and 3d domains comp meth appl mech engrg 188 2000 625 int j num meth engrg 57 2003 899 exponential shape functions in the longitudinal direction are used similarly as in comp meth appl mech engrg 140 1997 221 however arbitrary order of approximation in the tangential direction may be selected due to compatibility with the hp adaptive edge fe element discretization for maxwell s equations in bounded domains reported in comp meth appl mech engrg 152 1998 103 int j num meth engrg 53 2002 147 comp meth appl mech engrg 169 1999 331

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

the solution of a launch vehicle trajectory problem by an adaptive finite element method we describe a second order discontinuous galerkin finite element method for the solution of an optimal control problem for determining the trajectory of a launch vehicle we derive an a posteriori error estimate that is subsequently implemented as the basis for adaptive error control we demonstrate that the computational error estimate is reliable and accurate while the adaptive error control provides a significant gain in efficiency over uniform discretizations

macroscopic limit cycle via pure noise induced phase transitions bistability generated via a pure noise induced phase transition is reexamined from the view of bifurcations in macroscopic cumulant dynamics it allows an analytical study of the phase diagram in more general cases than previous methods in addition using this approach we investigate spatially extended systems with two degrees of freedom per site for this system the analytic solution of the stationary fokker planck equation is not available and a standard mean field approach cannot be used to find noise induced phase transitions a different approach based on cumulant dynamics predicts a noise induced phase transition through a hopf bifurcation leading to a macroscopic limit cycle motion which is confirmed by numerical simulation

hp optimization of finite element approximations analysis of the optimal mesh sequences in one dimension this paper addresses the problem of the construction of the optimal hp meshes for the singular solution x 12 in the one dimensional setting first a review of the asymptotic analysis of gui and babuska 1 is given and is used to characterize the optimal hp meshes which are geometrically graded toward the singular point the asymptotically optimal hp meshes are compared with the hp meshes which are constructed by employing the hp strategy of rachowicz et al 2 which employs the principle of maximum change of error per change in the number of degrees of freedom and the true optimal hp meshes which minimize the error for fixed number of degrees of freedom and which are determined by searching through the entire sequence of the admissible hp meshes it is shown that for the singular solution the meshes constructed by the hp strategy of rachowicz coincide with the true optimal meshes nevertheless there exist various solutions for which the algorithm which employs the principle of maximum change of error may not converge for this reason the hp strategy of rachowicz is combined with the classical feedback algorithm to obtain a robust hp strategy which always converges further it is shown that the asymptotically optimal hp meshes are very close to the true optimal hp meshes in the entire range of practical computation the effect of the accuracy of the error estimator on the optimality of the constructed hp meshes is also analyzed it is shown that unless the estimator is sufficiently accurate in the element adjacent to the singular point the meshes constructed by the adaptive hp strategy can differ substantially from the true optimal meshes

essential boundary conditions and multi point constraints in finite element analysis the application of boundary conditions and other constraints to the stiffness matrix and load vector is an integral part of any finite element code this process is usually trivial but can present difficulties to the point where some codes disallow certain combinations of commonly occurring boundary conditions a general approach to this problem is presented and applied to representative examples necessary and sufficient conditions are obtained under which constraints can be applied sequentially bounds are obtained for the condition number of the reduced system showing it is essentially the same as for the original problem

high order taylor galerkin and adaptive h p methods for second order hyperbolic systems application to elastodynamics a new family of high order taylor galerkin schemes is presented for the analysis of second order hyperbolic systems the use of such schemes within an adaptive h p finite element strategy for spatial approximation in two dimensional problems is described numerical results are presented to demonstrate performance of these methods

hierarchies of landau lifshitz bloch equations for nanomagnets a functional integral framework we propose a functional integral framework for the derivation of hierarchies of landau lifshitz bloch llb equations that describe the flow toward equilibrium of the first and second moments of the magnetization the short scale description is defined by the stochastic landau lifshitz gilbert equation under both markovian or non markovian noise and takes into account interaction terms that are of practical relevance depending on the interactions different hierarchies on the moments are obtained in the corresponding llb equations two closure ans atze are discussed and tested by numerical methods that are adapted to the symmetries of the problem our formalism provides a rigorous bridge between the atomistic spin dynamics simulations at short scales and micromagnetic descriptions at larger scales

critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling computational techniques within the population density function pdf framework have provided time saving alternatives to classical monte carlo simulations of neural network activity efficiency of the pdf method is lost as the underlying neuron model is made more realistic and the number of state variables increases in a detailed theoretical and computational study we elucidate strengths and weaknesses of dimension reduction by a particular moment closure method cai tao shelley mclaughlin 2004 cai tao rangan mclaughlin 2006 as applied to integrate and fire neurons that receive excitatory synaptic input only when the unitary postsynaptic conductance event has a single exponential time course the evolution equation for the pdf is a partial differential integral equation in two state variables voltage and excitatory conductance in the moment closure method one approximates the conditional kth centered moment of excitatory conductance given voltage by the corresponding unconditioned moment the result is a system of k coupled partial differential equations with one state variable voltage and k coupled ordinary differential equations moment closure at k 2 works well and at k 3 works even better in the regime of high dynamically varying synaptic input rates both closures break down at lower synaptic input rates phase plane analysis of the k 2 problem with typical parameters proves and reveals why no steady state solutions exist below a synaptic input rate that gives a firing rate of 59 s 1 in the full 2d problem closure at k 3 fails for similar reasons low firing rate solutions can be obtained only with parameters for the amplitude or kinetics or both of the unitary postsynaptic conductance event that are on the edge of the physiological range we conclude that this dimension reduction method gives ill posed problems for a wide range of physiological parameters and we suggest future directions

the h p and h p versions of the finite element method in 1 dimension part ii the error analysis of the h and h p versions the paper is the second in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimensional setting the first part 1 analyzed thep version the present one concentrates on theh andh p versions

application of hp adaptive be fe methods to elastic scattering abstract the investigation focuses on the development of hp adaptive methods for simulation of acoustical scattering problems the present formulation is based on steady state linear elasticity equations coupled with the burton miller integral equation representing the behavior of a surrounding acoustical medium the highlights of the proposed approach include a mathematically consistent variational formulation of the problem an hp adaptive fe be approximation based on standard galerkin methods an element by element solver a residual based a posteriori error estimate an hp adaptive strategy particular attention is given to asymptotic convergence and to a comparison of theory and numerical simulations

on the gaussian approximation for master equations we analyze the gaussian approximation as a method to obtain the first and second moments of a stochastic process described by a master equation we justify the use of this approximation with ideas coming from van kampen s expansion approach the fact that the probability distribution is gaussian at first order we analyze the scaling of the error with a large parameter of the system and compare it with van kampen s method our theoretical analysis and the study of several examples shows that the gaussian approximation turns out to be more accurate than van kampen s expansion at first order this could be specially important for problems involving stochastic processes in systems with a small number of particles

expectation and variation in multi period decisions multi period decisions are decisions which determine an individual s payoffs in several periods in the future this paper examines the theoretical foundations of the prevalent weighted average assumption more specifically we use a multi period interpretation of the famous ellsberg paradox in decision under uncertainty to show that in many cases of interest additively separable functionals in general and weighted average ones in particular do not seem appropriate for the representation of the decision maker s preferences we then suggest replacing the sure thing principle which may be used to axiomatize a weighted average functional by a weaker version of it using the weakened axiom in schmeidler s nonadditive measure model reinterpreted for the multi period context yields an axiomatization of a larger class of decision rules which are representable by a weighted average of the utility in each period und the utility variation between each two consecutive periods the weighted average assumption is a special case of the generalized model a case in which the decision maker is variation neutral similarly we define and characterize variation aversion and variation liking and show an example of the economic implications of these properties

free vibration analysis of membranes using the h p version of the finite element method abstract the h p version of the finite element method is applied to the vibration of membranes this is accomplished using a polynomially enriched triangular element new simple expressions of hierarchical c 0 shape functions for triangles are given in terms of the shifted legendre orthogonal polynomials the h p version of the finite element method marries both the concepts of the conventional h version and the p version the accuracy of the solution is sought by simultaneously refining the mesh and increasing the polynomial order in each element results of frequency calculations are found for triangular and l shaped membranes using a number of meshes and polynomial orders it is shown that the h p version of the finite element method is always convergent from above and produces a high accuracy with few degrees of freedom

aspects of an adaptive hp finite element method adaptive strategy conforming approximation and efficient solvers the main components needed for an adaptive hp version finite element algorithm are discussed an adaptive hp refinement strategy effective methods for constructing conforming hp approximations and efficient solvers for the large ill conditioned systems of linear equations together these provide the methodology for an effective adaptive hp version algorithm the presentation emphasizes the links between the differing components showing how the algorithms may be implemented efficiently in practice the main principles are illustrated by use of concrete examples so that a non expert may develop their own adaptive hp code the performance of the whole algorithm is illustrated for some representative problems taken from linear elasticity

locking free mixed hp finite element methods for curvilinear domains mixed hp finite element methods fems are robust with respect to poisson ratio locking when straight sided quadrilateral elements are used for the p version which uses fixed and generally large elements curved elements are necessary to represent curved domains accurately we show that mixed hp curved quadrilateral elements retain the robustness of their straight sided counterparts provided degenerate elements are avoided we also show that hp version triangles are robust we use these concepts to develop a locking free fem for geometrically nonlinear elasticity

adaptive finite element methods for hyperbolic systems with application to transient acoustics the solution of the hyperbolic systems of equations governing transient acoustics by adaptive finite elements and various finite difference time discretization schemes is addressed emphasis is placed on the use of a class of implicit runge kutta methods for temporal approximations

efficient preconditioning for the p version finite element method in two dimensions parallel preconditioners are formulated and analyzed for systems of equations arising from the p version finite element method applied to second order self adjoint elliptic boundary value problems in two dimensions by using new theoretical results for polynomial spaces it is proved that the condition number of the preconditioned system grows as log 2 p where p is the degree of the polynomial space numerical results are presented showing that the condition number indeed grows very slowly with p

a parallel adaptive strategy for hp finite element computations abstract in this work a new three step adaptive strategy for hp finite element discretizations is presented techniques for parallelizing these calculations on distributed memory multiprocessor computers are also developed numerical examples are used to demonstrate convergence properties and parallel efficiencies orders of magnitude reduction in computational efficiency over conventional methods are observed

on an adaptive monotonic convection diffusion flux discretization scheme abstract the present paper concerns numerical investigation of a two dimensional steady convection diffusion scalar equation specific attention is given to resolving spurious oscillations in a confined region of high gradient in smooth regions a high order accurate solution is desired while in regions containing a sharp gradient the strategy of applying a monotonic capturing scheme is preferred in this paper we model fluxes by means of a finite element model which has spaces spanned by legendre polynomials the propensity to yield an irreducible diagonal dominant stiffness matrix is an attribute of this finite element flux discretization scheme fundamental analyses are extensively conducted in this paper in two areas first by conducting a modified equation analysis we are led to confirm the consistency of the scheme both the stability and monotonicity of the solutions are also addressed a guide for judging whether the stiffness matrix can yield a monotonic solution is rooted in the theory of the m matrix this monotonicity study provides us with greater numerical insight into the importance of the chosen peclet number beyond its critical value oscillatory solutions are present in the flow this monotonic region is however fairly restricted it is this shortcoming which limits the finite element practioner s choices for fairly small grid size the scope of application using the proposed upwind scheme is thus rather small because monotonic solutions are prohibitively expensive to compute circumvention of such deficiency is accomplished by making modifications to the structured grid several test cases have been employed to examine the grid adaption technique for the treatment of advection terms in flow regions having sharp gradients considerable savings in computer storage and execution time are achieved by the employed unstructured grid setting

closure schemes in stochastic nonlinear dynamics a validation case study it is known that randomness in dynamical systems often leads to infinite hierarchies of coupled equations for relevant probabilistic quantities several closure methods for truncation of the hierarchies have been proposed in the literature in the present paper the performance of closure schemes for moment hierarchies is compared by using the well known nonlinear equation of overdamped oscillator with additive gaussian white noise in the case of bistable dynamics it is shown that the closure schemes can give incorrect results despite of their good convergence properties to overcome this deficiency new closure procedures are proposed they are based on using the hermite polynomials and their generalization

closing the hierarchy of moment equations in nonlinear dynamical systems the moment equations associated with the evolution of the probability density are known to form an infinite hierarchy of coupled equations in nonlinear dynamical systems in the present paper a systematic approach for closing this hierarchy is proposed based on the ansatz that in the long time limit there exist groups of moments varying on the same time scale the method is applied to a one dimensional vector field in the presence of noise and to two prototypes of chaotic behavior excellent agreement with numerical results is obtained special emphasis is placed on the role of symmetries and on the origin of the composite oscillations found for certain types of moments in the chaotic systems

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

estimation of computational homogenization error by explicit residual method in this paper a new method is introduced for the estimation of modeling error resulting from homogenization of elastic heterogeneous bodies the approach is similar to the well known explicit residual approximation error estimation it is proved that besides the residuum of the equilibrium equation and interelement traction jump also a difference of stress divergences as well as traction jump along the material interfaces contribute to the modeling error estimate moreover explicit specification and numerical evaluation of stability constants provide reasonable effectivity index of this error indicator selected numerical examples illustrate the promise of this approach therefore the proposed methodology is a computationally inexpensive option for the other methods of modeling error assessment

adaptive multiresolution antenna modeling using hierarchical mixed order tangential vector finite elements hierarchical mixed order tangential vector finite elements tvfes for tetrahedra are attractive for accurate and efficient finite element or hybrid finite element boundary integral fe bi simulation of complicated electromagnetic problems they provide versatility in the geometrical modeling of physical structures guarantee solutions free of spurious modes and allow a local increase of resolution by combination of mixed order tvfes of different orders within a computational domain regions with higher order mixed order tvfes can be selected a priori or be found adaptively via methods for a posteriori error estimation or indication this paper demonstrates the merits of various explicit residual methods for a posteriori error indication via adaptive refinement of fe bi solutions of electromagnetic radiation problems using hierarchical mixed order tvfes for tetrahedra

a new triangular and tetrahedral basis for high order hp finite element methods in this paper we describe the foundations of a new hierarchical modal basis suitable for high order hp finite element discretizations on unstructured meshes it is based on a generalized tensor product of mixed weight jacobi polynomials the generalized tensor product property leads to a low operation count with the use of sum factorization techniques variable p order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two and three dmensional computational domains employing standard meshes

a posteriori error estimators for second order elliptic systems part 2 an optimal order process for calculating self equilibrating fluxes abstract we present full details of an algorithm which can be used to obtain self equilibrating fluxes for arbitrary h p finite element approximations all of the calculations are performed on a local scale yielding an optimal order process o n where n is the number of elements in the mesh owing to the local nature of the process the algorithm is perfectly suited to parallel implementation the algorithm can be applied in both two and three dimensions for unstructured k irregular meshes and elements of non uniform degree finally it is shown that the implementation of the flux equilibration algorithm can take advantage of calculations already performed in obtaining the finite element approximation itself

a posteriori error estimators for second order elliptic systems part 1 theoretical foundations and a posteriori error analysis abstract this is the first in a series of two papers dealing with a posteriori error estimation for h p finite element approximation of second order elliptic systems in this paper we shall present the fundamental ideas behind the error estimator and the theoretical foundations for the method the second paper deals with the algorithmical details

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

an explicit a posteriori error indicator for electromagnetic finite element boundary integral analysis within the finite element method fem there is always an error associated with the solution quantifying this error is a very important aspect of fe analysis since it can increase confidence in a solution and can be used to drive adaptive refinements of the discretization optimizing the quality of the solution relative to the number of degrees of freedom and in turn reducing computational cost this paper presents an a posteriori error indicator for time harmonic electromagnetic analysis in three dimensions when using a galerkin fe approach with curl conforming elements and including a nonlocal boundary integral boundary condition as well as a dominant mode port boundary condition the error indicator is residual based of an explicit nature and constitutes a bound on an error energy measure the error indicator is evaluated by presenting results relating to common microwave engineering applications which shows that it can be used successfully to estimate the relative error distribution and that it can successfully serve within an adaptive algorithm

a hierarchical iterative procedure for the analysis of composite laminates a multilevel recursively defined preconditioner for use with the preconditioned conjugate gradient pcg algorithm is developed and applied to solve thin and thick laminated plate problems the preconditioner is constructed from a sequence of hierarchical vector spaces arising from the p version of the finite element fe method numerical studies have been conducted using quadrilateral elements for two dimensional problems up to p 8 and serendipity brick elements for three dimensional problems up to p 5 the efficiency of the iterative procedure is illustrated using bending response of laminated plates the effects of element span ratios convergence tolerance polynomial order material properties and plate span ratios on the number of iterations required for convergence has been investigated results indicate that the preconditioner developed herein can be used to produce an efficient iterative solver for two and three dimensional problems in structural mechanics

the multi level hp method for three dimensional problems dynamically changing high order mesh refinement with arbitrary hanging nodes abstract one main challenge of the h p version of the finite element method is the high implementational complexity of the method resulting from the added need of handling hanging nodes appropriately the multi level h p formulation recently introduced for two dimensional applications aims at alleviating these difficulties without compromising the approximation quality this is achieved by changing from the conventional refine by replacement approach to a refine by superposition idea the current work shows that the multi level h p approach can be extended naturally to three dimensional refinement without increasing the complexity of the rule set ensuring linear independence and compatibility of the shape functions in this way a three dimensional h p refinement scheme is formulated in which hanging nodes are avoided by definition this ease of complexity allows for a highly flexible discretization kernel featuring arbitrary irregular meshes and a continuous refinement and coarsening throughout the simulation runtime different numerical examples demonstrate that even in the presence of singularities this novel refinement scheme yields exponential convergence with respect to both the number of unknowns and the computational time it is further shown that the refinement scheme is able to capture complex solution features that demand for three dimensional refinement patterns the dynamic discretization properties of the approach are demonstrated by continuously refining and coarsening the mesh during the simulation runtime to keep the refinement zone local to a moving singularity finally it is shown that the high approximation power of the multi level h p scheme also carries over to curved geometries common in engineering practice without a significant detrimental effect on the conditioning of the stiffness matrix

the h p adaptive finite element method for the numerical simulation of compressible flow an h p adaptive finite element method is presented for the numerical simulation of compressible fluid flow in addition to refining the mesh in regions where the estimated error is large the polynomial degree of the elements is increased it is shown that by using high order polynomials highly accurate approximations are obtained to a regularized form of the equations governing viscous compressible flow

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

convergence and optimality of mathbf hp hp afem we design and analyze an adaptive hp finite element method mathbf hp hp afem in dimensions n 1 2 n 1 2 the algorithm consists of iterating two routines mathbf hp hp nearbest finds a near best hp approximation of the current discrete solution and data to a desired accuracy and reduce improves the discrete solution to a finer but comparable accuracy the former hinges on a recent algorithm by binev for adaptive hp approximation and acts as a coarsening step we prove convergence and instance optimality

a collection of 2d elliptic problems for testing adaptive grid refinement algorithms adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations pdes the development of new algorithms and computer codes for the solution of pdes usually involves the use of proof of concept test problems 2d elliptic problems are often used as the first test bed for new algorithms and codes this paper contains a set of twelve parametrized 2d elliptic test problems for adaptive grid refinement algorithms and codes the problems exhibit a variety of types of singularities near singularities and other difficulties

finite size fluctuations and stochastic resonance in globally coupled bistable systems the dynamics of a system formed by a finite number n of globally coupled bistable oscillators and driven by external forces is studied focusing on a global variable defined as the arithmetic mean of each oscillator variable several models based on truncation schemes of a hierarchy of stochastic equations for a set of fluctuating cumulant variables are presented this hierarchy is derived using it o stochastic calculus and the noise terms in it are treated using an asymptotic approximation valid for large n in addition a simplified one variable model based on an effective potential is also considered these models are tested in the framework of the phenomenon of stochastic resonance in turn they are used to explain in simple terms the very large gains recently observed in these finite systems

a systematic strategy for simultaneous adaptive hp finite element mesh modification using nonlinear programming abstract adaptive refinement usually involves refining or enriching a fraction of mesh elements by one level based on a cut off criterion requiring several costly intermediate solutions before a mesh that yields an acceptable solution is obtained we avoid this by formulating and solving the mesh design problem as a mathematical program our approach simultaneously modifies both mesh size h and local polynomial order p to yield an optimal mesh for a target error or given computational cost with gradients from local convergence rates constraints such as the one irregularity rule during mesh refinement are systematically incorporated in this formulation the design task leads to a mixed integer nonlinear program minlp that is relaxed to an nlp to reduce the computations for the nlp we employ simplified analytical gradients derived from initial mesh calculations finally we apply our method to three model problems showing that complex hp adaptive grids can be obtained directly from a uniform coarse grid a commercial optimization software minos b a murtagh m a saunders minos 5 4 user s guide technical report sol 83 20r stanford university stanford 1987 revised february 1995 was used as the nlp optimizer

a posteriori error estimation of h p finite element approximations of frictional contact problems abstract dynamic and static frictional contact problems are described using the normal compliance law on the contact boundary dynamic problems are recast into quasistatic problems by time discretization an a posteriori error estimator is developed for a nonlinear elliptic equation of corresponding static or quasistatic problems the a posteriori error estimator is applied to a frictionless case and extended to frictional contact problems also an adaptive strategy is introduced and h p finite element meshes are obtained through a procedure based on a priori and a posteriori error estimations numerical examples are given to support the theoretical results

prophlex an hp adaptive finite element kernel for solving coupled systems of partial differential equations in computational mechanics this paper presents an overview of the basic design and architecture of the prophlex hp adaptive finite element kernel prophlex was designed to be a commercial robust implementation of hp adaptivity driven by residual error estimation with the primary goal of being physics independent and computationally efficient on a wide array of computer hardware platforms prophlex can solve virtually any class of engineering problems which be may be mathematically formulated as a system of linear or nonlinear second order partial differential equations and associated boundary conditions it has been applied to compressible and incompressible fluid dynamics linear and nonlinear solid mechanics heat flow problems as well as semiconductor device simulation examples of prophlex customization for linear and nonlinear solid mechanics are presented

schwarz methods with local refinement for the p version finite element method in some applications the accuracy of the numerical solution of an elliptic problem needs to be increased only in certain parts of the domain in this paper local refinement is introduced for an overlapping additive schwarz algorithm for the p version finite element method both uniform and variable degree refinements are considered the resulting algorithm is highly parallel and scalable

a priori error estimations of hp finite element approximations for hierarchical models of plate and shell like structures a priori error estimates of hp finite element approximations for hierarchical models of the thin elastic plate and shell like structures are derived the error in the approximations of a hierarchical model consists of the modelling error due to assumptions on displacement or stress fields and the finite element approximation error of discretizations of such a model also non uniformity with respect to the thickness owing to numerical locking is included in the present error estimates numerical results are given supporting the theoretical results

the h p and h p versions of the finite element methods in 1 dimension part iii the adaptive h p version abstract the paper is the third and final part in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimension the first part analyzed the p version and the second part concentrated on the h and h p version this paper analyzes a theoretical frame of the adaptive h p version and based on it the authors provide concrete algorithm for the one dimensional problem it is proven that in the case that the solution has x sub alpha type singularity the adaptive algorithm give an exponential rate of convergence very close to the optimal one analyzed in the second part of the paper additional keyword error analysis

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

high order finite elements applied to the discrete boltzmann equation a discontinuous galerkin approach for solving the discrete boltzmann equation is presented allowing to compute approximate solutions for fluid flow problems based on a two dimensional high order finite element and an explicit euler time stepping scheme the d2q9 model is discretized and the results are compared to the exact solution of the navier stokes equation four numerical examples are considered including stationary and instationary problems with curved boundaries it is demonstrated that the proposed method allows to obtain the desired highly efficient exponential convergence copyright 2006 john wiley sons ltd

on weak residual error estimation a general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated some numerical results are given to show the effectiveness and efficiency of the estimators

foundations of intrinsic habit formation we provide theoretical foundations for several common nested representations of intrinsic linear habit formation our axiomatization introduces an intertemporal theory of weaning a decision maker from her habits using the device of compensation we clarify differences across specifications of the model provide measures of habit forming tendencies and suggest methods for axiomatizing time nonseparable preferences

an hp adaptive finite element method for electromagnetics part 3 a three dimensional infinite element for maxwell s equations this paper is a continuation of reference 26 cecot demkowicz and rachowicz computer methods in applied mechanics and engineering 2000 188 625 643 and describes an implementation of the infinite element for three dimensional time harmonic maxwell s equations proposed in reference 15 demkowicz and pal computer methods in applied mechanics and engineering 1998 164 77 94 the element is compatible with the hp finite element discretizations for maxwell s equations in bounded domains reported in references 16 18 computer methods in applied mechanics and engineering 1998 152 103 124 1999 169 331 344 2000 187 307 337 copyright 2003 john wiley sons ltd

sequentially continuous non monotonic choquet integrals schmeidler proc amer math soc 97 2 1986 255 261 established an integral representation theorem through the choquet integral for functionals satisfying monotonicity and a weaker condition than additivity namely comonotonic additivity murofushi sugeno machida fuzzy sets and systems 64 1994 73 86 generalize this representation to the case of bounded variation functionals omitting the monotonicity condition we give an alternative approach which is based on sequential continuity and tolerates non monotonicity this later condition is equivalent to s additivity in measure theory

optimal h p finite element methods abstract a theory of optimal petrov galerkin h p version finite element approximations is presented the optimal scheme is denned relative to a fine mesh solution space and relative to an arbitrary symmetric bilinear form the optimal method leads to a symmetric positive definite stiffness matrix which is independent of the coefficients of the given problem exhibits extra superconvergence properties and has a relative error that can be calculated exactly at each point in the problem domain various generalizations are also discussed including the connection of these methods with certain preconditioning schemes

hp version discontinuous galerkin methods for hyperbolic conservation laws a parallel adaptive strategy this paper describes a parallel algorithm based on discontinuous hp finite element approximations of linear scalar hyperbolic conservation laws the paper focuses on the development of an effective parallel adaptive strategy for such problems numerical experiments suggest that these techniques are highly parallelizable and exponentially convergent thereby yielding efficiency many times superior to conventional schemes for hyperbolic problems

solution of elastic scattering problems in linear acoustics using h p boundary element method abstract the work is a continuation of 1 a simplified 3d model problem reduced from the general class of elastic scattering problems is analyzed focusing attention on such practical details of the 3d implementation as a triangles based h p approximation and corresponding data structure integration of singular integrals and its effect on theoretically optimal rates of convergence a posteriori error estimation and sample adaptive strategies

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

time continuity and nonadditive expected utility information consisting of probabilities of some but possibly not all events induces an integral with respect to a probability specified on a subalgebra a decision maker evaluates the alternatives using only the available information and completely ignoring unavailable information assume now that the decision maker assesses the worth of a different lottery at each point in a discrete time assume also that each such lottery is preferred to some fixed alternative lottery now consider the situation where the sequence of lotteries converges in some sense if the limiting lottery is preferred to the fixed alternative then the preference order is referred to as time continuous this paper studies time continuity for two preference functionals the choquet integral and the integral with respect to a probability specified on a subalgebra the integral with respect to probability specified on a subalgebra is determined by the structure of the available information by relating it to the choquet integral we characterize the structure of available information that would yield time continuity

adaptive mesh generation for fluid mechanics problems in this study we give an in depth investigation of the adaptive procedures for compressible and incompressible flow problems many tests have been carried out with different error indicators for different problems these refinement procedures show that the interpolation error indicators are adequate to predict an accurate solution comparison between the gradient and curvature based error indicators show that the former yields better performance in most situations however we found that the latter indicators predict better motion in the recirculatory and unstable regions than the gradient based method many test cases have been presented for both compressible and incompressible flow problems in the later part of the paper we suggest a simple way of combining the curvature gradient indicators of different key variables and present an appropriate problem where it is applicable a wide coverage of relevant literature has also been presented in this paper copyright 2000 john wiley sons ltd