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

algorithm and architecture for a galois field multiplicative arithmetic processor we present a new algorithm for generic multiplicative computations of the form ab c in gf p sup m including multiplication inversion squaring and division the algorithm is based on solving a sequence of congruences that are derived from the theory of grobner bases in modules over the polynomial ring gf p x its corresponding hardware and software architectures can be successfully used in applications such as error control coding and cryptography we describe a versatile circuit associated with the algorithm for the most important case p 2 the same hardware can be used for a range of field sizes thus permitting applications in which different levels of error control or of security are required by different classes of user the operations listed are all performed by the hardware in the same number of clock cycles which prevents certain side channel attacks the loss in performance by having 2m iterations for multiplication is compensated by the full parameterization of the galois field and the ability to perform division and multiplication in parallel

characteristic local grid refinement techniques for reservoir flow problems using a sequential time marching approach we describe and implement effective solution strategies for the equations governing immiscible displacement in a porous medium by combining adaptive grid refinement with an operator splitting technique based on the modified method of characteristics the saturation equation is treated in a consistent and accurate way for the pressure equation we construct accurate piecewise linear velocity components from a piecewise linear pressure approximation

a petrov galerkin finite element method for variable coefficient fractional diffusion equations abstract fractional diffusion equations have found increasingly more applications in recent years but introduce new mathematical and numerical difficulties galerkin formulation which was proved to be coercive and well posed for fractional diffusion equations with a constant diffusivity coefficient may lose its coercivity for variable coefficient problems the corresponding finite element method fails to converge we utilize the discontinuous petrov galerkin dpg framework to develop a petrov galerkin finite element method for variable coefficient fractional diffusion equations we prove the well posedness and optimal order convergence of the petrov galerkin finite element method numerical examples are presented to verify the theoretical results

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

characteristic nonconforming finite element methods for advection dominated diffusion problems in this paper characteristic nonconforming finite element methods are studied for timedependent advection dominated diffusion problems the diffusion term in these problems is discretized using nonconforming finite elements and the temporal differentiation and advection terms are treated by characteristic tracking schemes various nonconforming finite elements and characteristic tracking schemes are studied stability and convergence properties of the characteristic nonconforming methods are obtained unconditionally stable results and error estimates of optimal order are established

stability and convergence of a finite element method for reactive transport in ground water an explicit finite element method is used to solve the linear convection diffusion reaction equations governing contaminant transport in ground water flowing through an adsorbing porous medium the use of discontinuous finite elements for the convective part of the equations combined with mixed finite elements for the diffusive part renders the method for the concentration solution which displays strong gradients trivially conservative and fully parallelizable we carry out a stability and convergence analysis in particular the method is proven to satisfy a maximum principle to be total variation bounded and to converge to the unique weak solution of the equations special attention is paid to the convective part of the equations numerical simulations are presented and discussed

finite volume solutions of convection diffusion test problems the cell vertex formulation of the finite volume method has been developed and widely used to model inviscid flows in aerodynamics more recently one of us has proposed an extension for viscous flows the purpose of the present paper is two fold first we have applied this scheme to a well known convection diffusion model problem involving flow round a 180 degrees bend which highlights some of the issues concerning the application of the boundary conditions in such cell based schemes the results are remarkably good when the boundary conditions are applied in an appropriate manner in our efforts to explain the high quality of the results we were led to a detailed analysis of the corresponding one dimensional problem our second purpose is thus to gather together various approaches to the analysis of this problem and to draw attention to the supra convergence phenomena enjoyed by the proposed methods 25 refs 9 figs 1 tab

a finite element method for an eikonal equation model of myocardial excitation wavefront propagation an efficient finite element method is developed to model the spreading of excitation in ventricular myocardium by treating the thin region of rapidly depolarizing tissue as a propagating wavefront the model is used to investigate excitation propagation in the full canine ventricular myocardium an eikonal curvature equation and an eikonal diffusion equation for excitation time are compared a petrov galerkin finite element method with cubic hermite elements is developed to solve the eikonal diffusion equation on a reasonably coarse mesh the oscillatory errors seen when using the galerkin weighted residual method with high mesh peclet numbers are avoided by supplementing the galerkin weights with c0 functions based on derivatives of the interpolation functions the ratio of the galerkin and supplementary weights is a function of the peclet number such that for one dimensional propagation the error in the solution is within a small constant factor of the optimal error achievable in the trial space an

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

on the numerical solution of non linear reservoir flow models with gravity based on a two phase fluid model for immiscible displacement in a porous medium we develop and analyse numerical solution techniques for certain non linear phenomena two different solution strategies for the treatment of gravity effects which are non trivial to model by existing solution techniques and may be of great influence in many practical flow situations are presented the solution procedures are based on an operator splitting technique combining the modified method of characteristics with finite element techniques and adaptive grid refinement

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

a robust petrov galerkin discretisation of convection diffusion equations a petrov galerkin discretisation is studied of an ultra weak variational formulation of the convection diffusion equation in a mixed form to arrive at an implementable method the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space to prevent that this latter space has to be taken increasingly large for vanishing diffusion a formulation is constructed that is well posed in the limit case of a pure transport problem numerical experiments show approximations that are very close to the best approximations to the solution from the trial space uniformly in the size of the diffusion term

uniform convergence of discontinuous finite element methods for singularly perturbed reaction diffusion problems abstract in this paper we study two families of discontinuous finite element methods for singularly perturbed reaction diffusion problems which contain multiple scales spanning the whole problem domain because of the small perturbation parameter to solve such a multiscale problem an anisotropic mesh is constructed this mesh uses finer mesh inside the small scale regions where the boundary layers are located than elsewhere large scale regions uniform optimal error estimates in the energy norm are obtained for both families on the anisotropical mesh

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

generation of canonical m sequences and dual bases an expression for the shift necessary to obtain a canonical m sequence is obtained in terms of the coefficients of the primitive polynomial that generates this m sequence this shift for obtaining a canonical m sequence is also directly related to the shift for obtaining the location of the span 100 0 in the m sequence in addition a recursion is developed for obtaining the dual basis elements of the basis expressing gf q sup m over gf q canonical m sequences and dual bases are important in the design of sequences for spread spectrum multiple access systems as well as in the design of general decoder structures using bit serial encoders

multidisciplinary interactions in energy and environmental modeling abstract mathematical models have been widely used to understand predict or optimize many complex physical processes from semiconductor or pharmaceutical design to large scale applications such as global weather models to astrophysics in particular simulation of environmental effects of air polution is extensive here we address the need for using similar models to understand the fate and transport of groundwater contaminants and to design in situ remediation strategies three basic problem areas must be addressed in the modeling and simulation of the flow of groundwater contamination one must first obtain an effective model to describe the complex fluid fluid and fluid rock interactions that control the transport of contaminants in groundwater this includes the problem of obtaining accurate reservoir descriptions at various length scales and modeling the effects of this heterogeneity in the reservoir simulators next one must develop accurate discretization techniques that retain the important physical properties of the continuous models finally one should develop efficient numerical solution algorithms that utilize the potential of the emerging computing architectures we will discuss recent advances in each of these three areas

quadratic streamline upwinding for finite element method solutions to 2 d convective transport problems abstract during the past decade finite element methods fems have been recognized to be more powerful tools in the solution of various flow problems as opposed to their predecessors finite difference methods on a fundamental level the fem involves approximations to the solution over a given trial space on a discrete mesh rather than finite difference approximations to the differential operators on a discrete mesh in this way the mechanics operators of the fem remain closer to the physical principles associated with a given problem in addition to this the fem incorporates boundary conditions very efficiently into the numerical formulation much current research is aimed at making these fems viable options for convection dominated flows in 1985 mizukami et al published two papers advancing the approach of streamline upwinding within linear elements for fems the purpose of this work is to demonstrate that streamline upwinding of higher ordered elements is a more accurate option the work goes on to show exactly how streamline upwinding effects the stability of iteratively solving the global algebraic system resulting from the fem

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

on some recent adventures into improved finite element cfd methods for convective transport abstract the quest continues for computational fluid dynamics cfd algorithms that are accurate and efficient for convection dominated applications including shocks travelling fronts and wall layers the boundary value optimal galerkin weak statement invariably requires manipulation either in the test space or in an augmented form for the conservation law system to handle the disruptive character introduced by the discretized first order convection term an incredible variety of methodologies have been derived and examined to address this issue in particular seeking achievement of accurate and monotone discrete approximate solutions in an efficient implementation the utk cfd research group continues its search on examining the breadth of approaches leading to development of a consistent encompassing theoretical statement exhibiting quality performance included herein are adventures into generalized taylor series lax wendroff methods characteristic euler flux resolutions sub grid embedded high degree lagrange bases with static condensation and assembled stencil fourier analysis optimization for finite element weak statement implementations for appropriate model problems including steady convection diffusion and pure unsteady convection and benchmark navier stokes definitions recent advances have lead to candidate accurate monotone methods with linear basis efficiency this contribution highlights the theoretical developments and presents quantitative documentation of achievable high quality solutions

constructing curves with high rank via symmetry for any g 1 we construct a curve of genus g defined over the rational function field of many variables q t 1 t n with rank at least 4 g 7 an immediate consequence is that there exists an infinite family of nonconstant curves of genus g over q with rank at least 4 g 7 which will improve the bound 3 g 7 which neron claimed in 1954

multiblock hybrid grid finite volume method to solve flow in irregular geometries in this work a finite volume based finite element method is suitably developed for solving incompressible flow and heat transfer on collocated hybrid grid topologies the method is generally applicable to arbitrarily shaped elements and orientations and thus challenges the potential to unify many of the different grid topologies into a single formulation the key point in this formulation is the correct estimation of the convective and diffusive fluxes at the cell faces using a novel physical influence scheme this scheme remarkably enhances the achieved solution accuracy it is shown that the extended formulation is robust enough to treat any combination of multiblock meshes with dual element shape employments in this regard the solution domain is broken up into a number of different multiblock arrangements in which each block is filled with only one type of finite element shape the combined grid can decrease the computational time and memory requirements and increase the numerical accuracy the results of the extended formulation are validated against different benchmark and other solutions the current results are in excellent agreement with the other solutions without exhibiting any disturbances around the block boundaries

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

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

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

efficient bit serial multiplication and the discrete time wiener hopf equation over finite fields discrete time wiener hopf equations dtwhes over finite fields are considered it is shown that solving the dtwhe is equivalent to performing division over finite fields the proof provides a new interpretation of the relationship between bit serial multiplication and dtwhes the interpretation enables bit serial multiplication over gf 2 sup m to be understood more easily as an example bit serial multiplication methods for multiplying any two elements that can be done without performing any transformation or with only a simple transformation of the bases are presented

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

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

the corrected operator splitting approach applied to a nonlinear advection diffusion problem so called corrected operator splitting methods are applied to a 1 d scalar advection diffusion equation of buckley leverett type with general initial data front tracking and a 2nd order godunov method are used to advance the solution in time diffusion is modelled by piece wise linear finite elements at each new time level to obtain correct structure of shock fronts independently of the size of the time step a dynamically defined residual flux term is grouped with diffusion different test problems are considered and the methods are compared with respect to accuracy and runtime finally we extend the corrected operator splitting to 2 d equations by means of dimensional splitting and we apply it to a buckley leverett type problem including gravitational effects

multiscale stabilization for convection dominated diffusion in heterogeneous media abstract we develop a petrov galerkin stabilization method for multiscale convection diffusion transport systems existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion which may not be sufficient to stabilize multiscale systems we seek a local reduced order model for this kind of multiscale transport problems and thus develop a systematic approach for finding reduced order approximations of the solution we start from a petrov galerkin framework using optimal weighting functions we introduce an auxiliary variable to a mixed formulation of the problem the auxiliary variable stands for the optimal weighting function the problem reduces to finding a test space a dimensionally reduced space for this auxiliary variable which guarantees that the error in the primal variable representing the solution is close to the projection error of the full solution on the dimensionally reduced space that approximates the solution to find the test space we reformulate some recent mixed generalized multiscale finite element methods we introduce snapshots and local spectral problems that appropriately define local weight and trial spaces in particular we use energy minimizing snapshots and local spectral decompositions in the natural norm associated with the auxiliary variable the resulting spectral decomposition adaptively identifies and builds the optimal multiscale space to stabilize the system we discuss the stability and its relation to the approximation property of the test space we design online basis functions which accelerate convergence in the test space and consequently improve stability we present several numerical examples and show that one needs a few test functions to achieve an error similar to the projection error in the primal variable irrespective of the peclet number

on bit serial multiplication and dual bases in gf 2 sup m the existence of certain types of dual bases in finite fields gf 2 sup m is discussed these special types of dual bases are needed for efficient implementation of generalized bit serial multiplication in gf 2 sup m in particular the question of choosing a polynomial basis of gf 2 sup m for example 1 alpha alpha sup 2 alpha sup 3 alpha sup m 1 such that the change of basis matrix from the dual basis to a scalar multiple of the original basis has as few 1 entries as possible is studied it was previously shown by m wang and i f blake 1990 that the optimal situation occurs when the minimal polynomial of alpha is an irreducible trinomial of degree m then an appropriate scalar multiple beta yields a change of basis matrix that is a permutation matrix a construction is presented that often yields bases where the change of basis matrix has low weight in the case where no irreducible trinomial of degree m exists a simple formula can be used to compute beta and the weight of the change of basis matrix given the minimal polynomial of alpha

adaptive anisotropic petrov galerkin methods for first order transport equations abstract this paper builds on recent developments of adaptive methods for linear transport equations based on certain stable variational formulations of petrov galerkin type the key issues can be summarized as follows the variational formulations allow us to employ meshes with cells of arbitrary aspect ratios we develop a refinement scheme generating highly anisotropic partitions that is inspired by shearlet systems we establish l 2 approximation rates for n term approximations from corresponding piecewise polynomials for certain compact cartoon classes of functions in contrast to earlier results in a curvelet or shearlet context the cartoon classes are concisely defined through certain characteristic parameters and the dependence of the approximation rates on these parameters is made explicit here the approximation rate results serve then as a benchmark for subsequent applications to adaptive galerkin solvers for transport equations we outline a new class of directionally adaptive petrov galerkin discretizations for such equations in numerical experiments the new algorithms track c 2 discontinuity curves stably and accurately and realize essentially optimal rates finally we treat parameter dependent transport problems which arise in kinetic models as well as in radiative transfer in heterogeneous media these problems feature propagation of singularities along curved characteristics precluding in particular fast marching methods based on ray tracing since now the solutions are functions of spatial variables and parameters one has to address the curse of dimensionality as for non adaptive schemes considered in grella and schwab 2011 and grella 2014 we show computationally for a model parametric transport problem in heterogeneous media in 2 1 dimension that sparse tensorization of the presently proposed directionally adaptive variational discretization scheme in physical space combined with hierarchic collocation in ordinate space can overcome the curse of dimensionality when approximating averaged bulk quantities

subspace correction methods for nonsymmetric parabolic problems abstract a convergence proof for general subspace correction techniques applied to an abstract nonsymmetric parabolic equation is given one of the main concerns is to give a unified convergence analysis for domain decomposition and multigrid methods for parabolic problems the analysis is also valid for nonsymmetric problems for second order parabolic problems the convection can dominate the diffusion the algorithms can be applied to domain decomposition methods with or without the coarse mesh for applications to multigrid methods the coarsest mesh does not need to be very coarse a relation between the coarse mesh size and the time step is needed to get a convergence rate independent of the mesh the number of iterations at each time step for the algorithms is also estimated some numerical experiments are presented for domain decomposition methods with minimum overlap which support the theoretical predictions the algorithms are able to capture the sharp traveling shocks for convection dominated problems

efficient implementation of high order methods for the advection diffusion equation abstract a new approach to designing high order defined here to exceed third accurate methods has been developed and tested for a linear advection diffusion equation in one and two dimensions the systematic construction of progressively higher order spatial approximations is achieved via a modified equation analysis which allows one to determine the computational stencil coefficients appropriate for a desired accuracy order a distinguishing desirable property of the developed method is solution matrix bandwidth containment i e bandwidth always remains equal to that of the second order discretization numerical simulations compare performance of the developed fourth and sixth order methods to that of the linear and bilinear basis galerkin weak statement formulations in one and two dimensions respectively uniform mesh refinement convergence results confirm the order of truncation error for each method high order approximations are shown to require significantly fewer nodes to accurately resolve solution gradients for convection dominated problems

extended variational formulations and f e models for nonlinear non conservative mechanical problems abstract for non conservative mechanical systems classical variational principles do not hold true however recent results on the so called inverse problem of calculus of variations lead to extended variational formulations which can be provided even for non conservative systems in this contribution f e models derived from extended variational principles are discussed in a significant case of an illconditioned balance equation it appears that extended mixed variational formulations could improve the numerical performance of f e methods for this class of problems

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

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

discontinuous petrov galerkin method with optimal test functions for thin body problems in solid mechanics we study the applicability of the discontinuous petrov galerkin dpg variational framework for thin body problems in structural mechanics our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate local computation of the corresponding optimal test functions in the timoshenko beam problem the proposed method is shown to provide the best approximation in an energy type norm which is equivalent to the l2 norm for all the unknowns uniformly with respect to the thickness parameter the same formulation remains valid also for the asymptotic euler bernoulli solution as another one dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations in particular we derive a special norm for the test space which leads to a robust method in terms of the shell thickness finally we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp mesh for resolving the boundary layer

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 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

an adaptive characteristic petrov galerkin finite element method for convection dominated linear and nonlinear parabolic problems in one space variable a new adaptive finite element method for convection dominated problems is presented a special feature of the method is that it is based on a petrov galerkin scheme for spatial approximation at a typical time step which employs test functions chosen so that the approximate solution coincides with the exact solution at the nodes of finite element grid this procedure makes possible the derivation of truly local a posteriori error estimates and a very effective solver numerical examples are discussed which illustrate the efficiency and effectively of the method

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

a space time accurate method for solving solute transport problems solute transport studies frequently rely on numerical solutions of the classical advection diffusion equation unfortunately solutions obtained with traditional finite difference and finite element techniques typically exhibit spurious damping or oscillation when advection dominates recently developed variants of these techniques such as the finite analytic method chen and li 1979 chen and chen 1984 and the optimal test function method celia et al 1989a b c perform well for steady state problems extensions of these methods to the transient case have however not been successful primarily because of inadequate approximations of the temporal derivative the new numerical method proposed in this paper avoids this difficulty by taking the laplace transform of the transient equation the transformed expression behaves like a steady state advection diffusion equation with a first order decay term this expression can be solved with either the finite analytic or optimal test function method and the time dependence recovered with an efficient inverse laplace transform algorithm the result is an accurate and robust transient solution which performs well over a very wide range of peclet numbers we demonstrate this approach by applying the finite analytic method to a laplace transformed one dimensional model problem a comparison with other competing techniques shows that good approximations are required in both space and time in order to obtain accurate solutions to advection dominated problems a good space approximation combined with a poor temporal approximation or vice versa does not give satisfactory results the method we propose provides a balanced space time approximation which works very well for one dimensional problems extensions to multiple dimensions are conceptually straightforward and briefly discussed

breaking spaces and forms for the dpg method and applications including maxwell equations discontinuous petrov galerkin dpg methods are made easily implementable using broken test spaces i e spaces of functions with no continuity constraints across mesh element interfaces broken spaces derivable from a standard exact sequence of first order unbroken sobolev spaces are of particular interest a characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces this technique is used to provide a complete error analysis of dpg methods for maxwell equations with perfect electric boundary conditions the technique also permits considerable simplifications of previous analyses of dpg methods for other equations reliability and efficiency estimates for an error indicator also follow finally the equivalence of stability for various formulations of the same maxwell problem is proved including the strong form the ultraweak form and various forms in between

on ranks of twists of elliptic curves and power free values of binary forms let e be an elliptic curve over q by the rank of e we shall mean the rank of the group of rational points of e mestre 31 improving on the work of neron 34 cf 13 39 and 46 has shown that there is an infinite family of elliptic curves over q with rank at least 12 however computational work see 3 4 7 and 47 suggests that a typical elliptic curve will have much smaller rank with curves of rank 0 or 1 being predominant indeed brumer 6 has proved subject to the birch swinnerton dyer conjecture the shimura taniyama weil conjecture and the generalized riemann hypothesis that the average rank of an elliptic curve ordered according to its faltings height is at most 2 3 in this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over q that is we shall restrict our attention to families of elliptic curves defined over q which are isomorphic over c there are families of quadratic cubic quartic and sextic twists see for example proposition 5 4 of chapter x of 42 let e be an elliptic curve over q with weierstrass equation y2 x ax b and for any non zero integer d let ed 2 3 denote a quadratic twist of e given by the equation dy2 x ax b let r d denote the rank of ed note that if d1 and d2 are non zero integers then ed is isomorphic to ed over q if and only if d1 d2 is the square of a rational number subject to the conjectures of birch and swinnerton dyer and of shimura taniyama and weil goldfeld 14 conjectured in 1979 that

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

a quasi optimal petrov galerkin method for helmholtz problem a petrov galerkin finite element formulation is introduced for helmholtz problem in two dimensions using polynomial weighting functions at each node of the mesh a global basis function for the weighting space is obtained adding to the bilinear c 0 lagrangian weighting function linear combinations of polynomial bubbles defined on a macroelement containing this node quasi optimal weighting functions with the same support of the corresponding global test functions are obtained after computing the coefficients of these linear combinations attending to optimality criteria this is done numerically through a simple preprocessing technique naturally applied to non uniform and unstructured meshes in particular for uniform mesh a quasi optimal interior stencil of the same order of the quasi stabilized finite element method stencil derived by babuska et al comput methods appl mech engrg 1995 128 325 359 is obtained numerical results are presented illustrating the great stability and accuracy of this formulation with uniform and non uniform meshes

global uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems higher order elements abstract in this paper we develop a general higher order finite element method for solving singularly perturbed elliptic linear and quasilinear problems in two space dimensions we prove that a quasioptimal global uniform convergence rate of 0 n x m 1 in m 1 n x n v m 1 in m 1 n v in l 2 norm is obtained for a reaction diffusion model by using the m th order m 2 tensor product element thus answering some open problems posed by roos in h g roos layer adapted grids for singular perturbation problems z angew math mech 78 5 1998 291 309 and h g ross m stynes and l tobiska numerical methods for singularly perturbed differential equations springer verlag berlin 1996 278 here n x and n v are the number of partitions in the x and y directions respectively numerical results are provided supporting our theoretical analysis

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

non local and semi local optimal weighting functions for symmetric problems involving a small parameter the method of optimal weighting functions for symmetric problems is described in a general form it is based on a petrov galerkin formulation in which the best approximation property and other mathematical features are achieved for a chosen norm different from the original energy norm of the problem the nonlocality of the weighting functions is shown to have only a minor effect on the efficiency of the method although a localization scheme is also suggested the method is applied to a one and two dimensional singular perturbation problems as well as to a cylindrical shell problem

eulerian lagrangian localized adjoint methods for a nonlinear advection diffusion equation eulerian lagrangian localized adjoint methods ellam are developed for the non linear buckley leverett equation which is characterized by degenerate diffusion and sharpening near shock solutions the ellam methodology employs space time finite elements with edges oriented along flow paths and space time test functions that satisfy a local adjoint condition this combination extends eulerian lagrangian concepts in a systematic mass conservative fashion to problems with general boundary conditions various kinds of boundary conditions are considered and a local time stepping procedure is developed for a no flow outlet condition that leads to a boundary layer numerical experiments illustrate the potential of these methods

on a penalty finite element cfd algorithm for high speed flow abstract the expanding interest in finite element discrete approximation techniques applied to high speed flow prediction has engendered derivation of many distinct theoretical statements quantization of algorithm robustness has focused on accuracy assessment for non smooth solutions i e those containing shocks and or contact discontinuities to reduced forms of the navier stokes equations specifically the euler equation system and the transonic potential flow equation this paper presents derivation of a penalty finite element algorithm applicable to both problem statements with penalty functional derived as a reduced form of a taylor galerkin weak statement numerical results for shocked transonic and euler solutions are cited to document aspects of accuracy convergence and stability

multiscale stabilization for convection diffusion equations with heterogeneous velocity and diffusion coefficients abstract we present a new stabilization technique for multiscale convection diffusion problems stabilization for these problems has been a challenging task especially for the case with high peclet numbers our method is based on a constraint energy minimization idea and the discontinuous petrov galerkin formulation in particular the test functions are constructed by minimizing an appropriate energy subject to certain orthogonality conditions and are related to the trial space the resulting test functions have a localization property and can therefore be computed locally we will prove the stability and present several numerical results our numerical results confirm that our test space gives a good stability in the sense that the solution error is close to the best approximation error

a variational multiscale a posteriori error estimation method for mixed form of nearly incompressible elasticity this paper presents an error estimation framework for a mixed displacement pressure finite element method for nearly incompressible elasticity the proposed method is based on variational multiscale vms concepts wherein the displacement field is decomposed into coarse scales that can be resolved by a given finite element mesh and fine scales that are beyond the resolution capacity of the mesh variational projection of fine scales onto the coarse scale space via variational embedding of the fine scale solution into the coarse scale formulation leads to the stabilized method with two major attributes first it is free of volumetric locking and second it accommodates arbitrary combinations of interpolation functions for the displacement and pressure fields this vms based stabilized method is equipped with naturally derived error estimators and offers various options for numerical computation of the error specifically two error estimators are explored the first method employs an element based strategy and a representation of error via a fine scale error equation defined over element interiors which is evaluated by a direct post solution evaluation this quantity when combined with the global pollution error results in a simple explicit error estimator the second method involves solving the fine scale error equation through localization to overlapping patches spread across the domain thereby leading to an implicit calculation of the local error this implicit calculation when combined with the global pollution error results in an implicit error estimator the performance of the stabilized method and the error estimators is investigated through numerical convergence tests conducted for two model problems on uniform and distorted meshes the sharpness and robustness of the estimators is shown to be consistent across the test cases employed

spatial filter averaging approach of probabilistic method to linear second order partial differential equations of the parabolic type a backward in time probabilistic method with spatial filter averaging is presented to solve linear second order partial differential equations of the parabolic type an advantage of this methodology is that while forward methods are subject to region with loss of density of particles and hence loss of spatial resolution of the solution the solution given by backward methods is given on any desired grid however traditional backward time probabilistic method using monte carlo averaging are computationally expensive we prove a convergence result and present several examples the method leads to important improvement in computational efficiency and is expected to perform well to solve high dimensional problems where a solution is needed on a large grid

spectral element simulations of interactive particles in a fluid abstract this paper presents three dimensional simulations of interactive particles in a fluid the originality of this paper is the integrated eulerian lagrangian algorithms implemented with high order methods in both space and time for numerical solutions to navier stokes equations with additional source terms implementation of fourth order stiffly stable schemes was tested for time integration before they were used in computations in order to provide the h p type mesh refinement for extra flexibility to achieve higher spatial resolution a modal spectral element method was used to solve governing equations in three dimensions another originality is the efficiency in handling moving particles without adaptive or moving mesh simulation results were validated with experimental data and verified with exact solutions in addition numerical results were also compared with solutions at higher resolutions and good agreement was accomplished results indicate that algorithms and implementations are accurate and appropriate for investigating three dimensional interactive particles in a fluid involving many moving objects

an infinite family of elliptic curves over q with large rank via neron s method in his famous article n1 n6ron has given a method for constructing an infinite family of elliptic curves over q with rank at least 11 by studying this from the viewpoint of mordell weil lattices is 1 s 2 s 3 we find that n6ron s method gives a complete algorithm for such a construction in particular we can write down an explicit numerical example n6ron s idea is a very ingeneous beautiful combination of some deep results in algebraic geometry and number theory the former is based on the theory of del pezzo surfaces cf m dp and the latter is the specialization argument originated by n6ron which has been strengthened by silverman and tate cf sil t another ingredient implicit in in1 seems to be the idea leading to the kodaira n6ron model an elliptic surface of an elliptic curve over a function field which was to be developed later in n2 our contribution if any will be

automatically stable discontinuous petrov galerkin methods for stationary transport problems quasi optimal test space norm we investigate the application of the discontinuous petrov galerkin dpg finite element framework to stationary convection diffusion problems in particular we demonstrate how the quasi optimal test space norm improves the robustness of the dpg method with respect to vanishing diffusion we numerically compare coarse mesh accuracy of the approximation when using the quasi optimal norm the standard norm and the weighted norm our results show that the quasi optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions we address the problems associated to the resolution of the optimal test functions with respect to the quasi optimal norm by studying their convergence numerically in order to facilitate understanding of the method we also include a detailed explanation of the methodology from the algorithmic point of view

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

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

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100