the h text p version of the finite element method for domains with curved boundaries this paper analyzes the convergence of the h text p version of the finite element method for elliptic problems on curved domains it proves that when the boundary of the domain is piecewise analytic boundary conditions are piecewise analytic and the sides of the elements are analytic the h text p version has an exponential rate of convergence numerical computations are included the paper generalizes the results previously obtained by the authors the h text p version of finite element method part 1 the basic approximation results comp mech 1 1986 pp 21 41 the h text p version of finite element method part 2 general results and applications comp mech 1 1986 pp 203 220

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

hp dgfem for second order elliptic problems in polyhedra ii exponential convergence the goal of this paper is to establish exponential convergence of hp version interior penalty ip discontinuous galerkin dg finite element methods for the numerical approximation of linear second order elliptic boundary value problems with homogeneous dirichlet boundary conditions and piecewise analytic data in three dimensional polyhedral domains more precisely we shall analyze the convergence of the hp ip dg methods considered in d schotzau c schwab t p wihler siam j numer anal 51 2013 pp 1610 1633 based on axiparallel sigma geometric anisotropic meshes and bm s linear anisotropic polynomial degree distributions

an a posteriori error estimator for the p and hp versions of the finite element method an a posteriori error estimator is proposed in this paper for the p and hp versions of the finite element method in two dimensional linear elastostatic problems the local error estimator consists in an enhancement of an error indicator proposed by bertoti and szabo int j numer meth engng 1998 42 561 587 which is based on the minimum complementary energy principle in order to obtain the local error estimate this error indicator is corrected by a factor which depends only on the polynomial degree of the element the proposed error estimator shows a good effectivity index in meshes with uniform and non uniform polynomial distributions especially when the global error is estimated furthermore the local error estimator is reliable enough to guide p and hp adaptive refinement strategies

on the computational complexity of ising spin glass models in a spin glass with ising spins the problems of computing the magnetic partition function and finding a ground state are studied in a finite two dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice in contrast to this fact the same problems are shown to belong to the class of np hard problems both in the two dimensional case within a magnetic field and in the three dimensional case np hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it

a fully automatic hp adaptivity for elliptic pdes in three dimensions abstract we present an algorithm that determines an optimal refinement strategy for a given finite element mesh the algorithm automatically selects optimal possibly anisotropic h refinements and distribution of also anisotropic order of approximation p the local selection between h refinement and p enrichment is made to decrease the projection based interpolation error of a reference fine grid solution with maximum rate with respect to the number of local degrees of freedom dof added a reconciliation step maintains the 1 irregularity of the mesh when applied iteratively the algorithm delivers exponential convergence in the energy norm for model elliptic problems with singular or nearly singular solutions

stability of the inversion of 3d fixed frequency scattering data abstract numerical methods for solving 3d inverse scattering problems with fixed energy data are described exact inversion formulas are given stability of the inversion methods based on these formulas is investigated an explicit formula is given for construction of the dirichlet to neumann map from the scattering amplitude this formula is used as a basis for another numerical approach to the inverse scattering problem several open problems are formulated

performance of the h p version of the finite element method with various elements the paper addresses the performance of square elements of type q p and q p q p and q p are elements of degree p analogous to the well known 9 and 8 noded elements for p 2 the performance is analysed theoretically for the class of analytic functions numerical experiments confirm the conclusions drawn from the theory the computational complexity of a solution algorithm is studied using timings of the computation on an alliant fx 8 computer the data show that high order elements are very effective for the considered class of solutions

theh p version of the finite element method for elliptic equations of order2m the problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data it has been shown in 3 4 5 that the solutions of the second order and fourth order equations belong to the spacesb 1 where the weighted sobolev norms of thek th derivatives are bounded bycd k l k l k l l 2 wherec andd are constants independent ofk in this case theh p version of the finite element method leads to an exponential rate of convergence measured in the energy norm 6 12 13 theh p version was implemented in the code probe1 18 and has been very successfully used in the industry we will discuss in this paper the generalization of these results for problems of order2m we will show also that the exponential rate can be achieved if the exact solution belongs to the spacesb 1 where the weighted sobolev norm of thek th derivatives is bounded bycd k l k l k l m 1 c andd are independent ofk in addition if the data is piecewise analytic then in fact the exact solution belongs to the spacesb m 1 problems of this type are related obviously to many engineering problems such as problems of plates and shells and are also important in connection with well known locking problems

3d inverse scattering abstract we give a self contained presentation of the method developed by r newton for solving the 3d inverse scattering problem and give an analysis of what is proved and what is not proved in this theory

hp adaptive finite elements in electromagnetics 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 curvilinear elements is presented the proposed method is illustrated and verified by a series of 2d experiments which include elements with curved boundaries and nonhomogeneous media

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

parallel space time hp adaptive discretization scheme for parabolic problems abstract we introduce an integration scheme for parabolic problems our parallelizable method uses adaptive h p finite elements in space and finite differences in time the strategy can also be combined with a classical finite element method parallelization technique based on domain decomposition we verified the performance of our method against two different benchmarks in both two dimensional model problem on an l shaped domain and three dimensional pennes bioheat equation settings results show a significant speedup in computational time when compared with the sequential version of the algorithm moreover we develop a mathematical framework to analyze similar schemes which include h p spatial adaptivity our framework describes error propagation rigorously and as such allows to analyze convergence properties of these mixed methods

weighted analytic regularity in polyhedra we explain a simple strategy to establish analytic regularity for solutions of second order linear elliptic boundary value problems the abstract framework presented here helps to understand the proof of analytic regularity in polyhedral domains given in the authors paper in m costabel m dauge s nicaise analytic regularity for linear elliptic systems in polygons and polyhedra math models methods appl sci 22 8 2012 we illustrate this strategy by considering problems set in smooth domains in corner domains and in polyhedra

computational experience with an interior point cutting plane algorithm there has been a great deal of success in the last 20 years with the use of cutting plane algorithms to solve specialized integer programming problems generally these algorithms work by solving a sequence of linear programming relaxations of the integer programming problem and they use the simplex algorithm to solve the relaxations in this paper we describe experiments using a predictor corrector interior point method to solve the relaxations for some problems the interior point code requires considerably less time than a simplex based cutting plane algorithm

discontinuous galerkin methods for the helmholtz equation with large wave number this paper develops and analyzes some interior penalty discontinuous galerkin ipdg methods using piecewise linear polynomials for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions it is proved that the proposed discontinuous galerkin methods are stable hence well posed without any mesh constraint for each fixed wave number k optimal order with respect to h error estimate in the broken h 1 norm and suboptimal order estimate in the l 2 norm are derived without any mesh constraint the latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime i e k 2 h gtrsim 1 numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect with respect to k in the error bounds the novelties of the proposed ipdg methods include the following first the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives second the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters since the helmholtz problem is a non hermitian and indefinite linear problem as expected the crucial and the most difficult part of the whole analysis is to establish the stability estimates i e a priori estimates for the numerical solutions to this end the cruxes of our analysis are to establish and to make use of a local version of the rellich identity for the laplacian and to mimic the stability analysis for the pde solutions given in p cummings analysis of finite element based numerical methods for acoustic waves elastic waves and fluid solid interactions in the frequency domain ph d thesis the university of tennessee knoxville tn 2001 p cummings and x feng math models methods appl sci 16 2006 pp 139 160 u hetmaniuk commun math sci 5 2007 pp 665 678

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

improved error estimation of dynamic finite element methods for second order parabolic equations dynamic finite element schemes are analyzed for second order parabolic problems these schemes permit different finite element spaces at different time levels in order to efficiently capture time changing localized phenomena such as moving sharp fronts or layers the dynamical change of grids and interpolation polynomials is necessary and essential to many large scale transient problems standard characteristic and mixed finite element methods with dynamic function spaces are considered for linear and nonlinear problems in a unified framework with improved a priori error estimates and convergence results

regularity and numerical solution of eigenvalue problems with piecewise analytic data this paper discusses the regularity of the eigenfunctions of eigenvalue problems with piecewise analytic data and the approximation of the eigenvalues and eigenfunctions of such problems a detailed and systematic numerical study of these approximations is presented together with an analysis of the numerical results in light of the theoretical results the specific aim is to assess the reliability of the theoretical results which are of an asymptotic nature as a guide to practical computations which may take place in the pre asymptotic phase and to look for characteristic features of the numerical results that are not completely explained by known theoretical results

pml enhanced with a self adaptive goal oriented hp finite element method simulation of through casing borehole resistivity measurements we describe the application of a perfectly matched layer pml combined with a self adaptive goal oriented hp finite element fe method to the simulation of borehole resistivity measurements the adaptive refinements and fast convergence of the self adaptive hp fe method enhance the performance of the pml thereby enabling the accurate and efficient truncation of the computational domain in open domain problems we apply this method to the simulation of axisymmetric through casing resistivity measurements acquired in a borehole environment that are typically used for the assessment of rock formation properties our numerical results confirm the accuracy and efficiency of our method and provide evidence of highly accurate and reliable simulations of borehole logging measurements in the presence of a conductive steel casing and material contrast of fourteen orders of magnitude in electrical conductivity moreover the combination of adaptivity and pml enables us to significantly reduce the size of the computational domain with substantial savings in computer time and memory

novel order parameter to describe the critical behavior of ising spin glass models abstract a novel order parameter for spin glasses is defined based on topological criteria and with a clear physical interpretation is first investigated for well known magnetic systems and then applied to the edwards anderson j model on a square lattice comparing its properties with the usual q order parameter finite size scaling procedures are performed results and analyses based on confirm a zero temperature phase transition and allow to identify the low temperature phase the advantages of are brought out and its physical meaning is established

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

simulation of wireline sonic logging measurements acquired with borehole eccentered tools using a high order adaptive finite element method the paper introduces a high order adaptive finite element method for simulation of sonic measurements acquired with borehole eccentered logging instruments the resulting frequency domain based algorithm combines a fourier series expansion in one spatial dimension with a two dimensional high order adaptive finite element method fem and incorporates a perfectly matched layer pml for truncation of the computational domain the simulation method was verified for various model problems including a comparison to a semi analytical solution developed specifically for this purpose numerical results indicate that for a wireline sonic tool operating in a fast formation the main propagation modes are insensitive to the distance from the center of the tool to the center of the borehole eccentricity distance however new flexural modes arise with an increase in eccentricity distance in soft formations we identify a new dipole tool mode which arises as a result of tool eccentricity

multidimensional nonlinear evolution equations and inverse scattering abstract in this paper we will review some recent work done in the field of integrable nonlinear evolution equations and inverse scattering we will concentrate on the basic underlying areas and refer interested readers to suitable references for complete details specifically background material can be found in various texts on this subject e g 1 by ablowitz and segur more recent references will be given as necessary the outline of the paper is as follows 1 1 introductory remarks 2 2 a discussion of two separate but related issues namely a solving certain nonlinear evolutions equations in infinite space and b inverse scattering these are important problems having many physical applications moreover they are related to each other by what we refer to as the inverse scattering transform ist 3 3 at the end of the paper we will make some remarks on the possibility of solving nonlinear evolution equations in high dimensions i e equations with more than two spatial and one time variable by using the ist method as we now understand it

the h p version of the coupling of finite element and boundary element methods for transmission problems in polyhedral domains this paper analyzes the rate of convergence of the h p version of the coupling of the finite element and boundary element method for transmission problems with a linear differential operator with variable coefficients in a bounded polyhedral domain omega 1 and with constant coefficients in the exterior domain omega 2 bbb r 3 backslash overline omega 1 this procedure uses the variational formulation of the differential equation in omega 1 and involves integral operators on the interface between omega 1 and omega 2 the finite elements are used to obtain approximate solutions of the differential equation in omega 1 and the boundary elements are used to obtain approximate solutions of the integral equations for given piecewise analytic data we show that the galerkin solution of this coupling procedure converges exponentially fast in the energy norm if the h p version is used both for finite elements and boundary elements

robustness of isogeometric structural discretizations under severe mesh distortion abstract this paper investigates higher order and higher continuity functions in isogeometric structural analysis under distortion of the control and physical meshes first the concepts behind isogeometric analysis are briefly reviewed the effect of perturbing control points upon the geometrical mapping is observed in the one dimensional setting the concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy a circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models the fillet patch is tested with two problems of linear elasticity and is shown to be accurate the investigation is further extended to the setting of large deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed f projection method

approximation properties of the h p version of the finite element method abstract the paper surveys the approximation and convergence properties of the h p version of the finite element method the basic theoretical results together with the main ideas of the proofs numerical examples are given

regularity of the solutions for elliptic problems on nonsmooth domains in r3 part 2 regularity in neighborhoods of edges this paper is the second in a series of three devoted to the analysis of the regularity of solutions of elliptic problems on nonsmooth domains in 3 the present paper concentrates on the regularity of solutions of the poisson equation in neighbourhoods of edges of a polyhedral domain in the framework of the weighted sobolev spaces and countably normed spaces these results can be generalised to elliptic problems arising from mechanics and engineering for instance the elasticity problem on polyhedral domains hence the results are not only important to understand comprehensively the qualitative and quantitative aspects of the behaviours of the solution and its derivatives of all orders in neighbourhoods of edges but also essential to design an effective computation and analyse the optimal convergence of the finite elements solutions for these problems

an hp finite element adaptive scheme to solve the laplace model for fluid solid vibrations abstract in this paper we introduce an hp finite element method to solve a two dimensional fluid structure spectral problem this problem arises from the computation of the vibration modes of a bundle of parallel tubes immersed in an incompressible fluid we prove the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues we define an a posteriori error estimator of the residual type which can be computed locally from the approximate eigenpair we show its reliability and efficiency by proving that the estimator is equivalent to the energy norm of the error up to higher order terms the equivalence constant of the efficiency estimate being suboptimal in that it depends on the polynomial degree we present an hp adaptive algorithm and several numerical tests which show the performance of the scheme including some numerical evidence of exponential convergence

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

evidence of exponential convergence in the computation of maxwell eigenvalues abstract this paper describes a finite element approach for the approximation of maxwell eigenvalues for three dimensional cavity problems hierarchical h 1 and h curl conforming discretisations are employed in conjunction with a technique that reduces the size of the problem to be solved numerical results are given on both polyhedral and curvilinear domains these results indicate that exponential rates of convergence can be obtained with the appropriate combination of h and p refinements

inverse scattering on half line the theory of inverse scattering on half line has been presented in the literature we give a new approach to the theory and analyze its logic in particular we prove that the potential reconstructed from the given scattering data generates the scattering data identical with the original data that the scattering data obtained from the given potential generate the potential which is identical with the original one that the normalizing constants can be put in a bijective correspondence to the parameters which define the general solution to the marchenko equation for x 0 we give a new characterization of the scattering data this characterization does not involve assumptions about the number of solutions to some integral equations in contrast with the known one applications of the theory are given necessary and sufficient conditions on the scattering data for q to be compactly supported or for q to be in l2 existence of a q c0 0 r which produces infinitely many purely imaginary resonances etc

a two dimensional self adaptive hp finite element method for the characterization of waveguide discontinuities part i energy norm based automatic hp adaptivity abstract the accurate analysis and characterization of waveguide discontinuities is an important issue in microwave engineering in some cases the discontinuities are an unavoidable result of mechanical or electric transitions which effects have to be minimized in other cases the discontinuities are deliberately introduced into the waveguide to perform a certain electric function this is the first of two papers analyzing different types of rectangular waveguide discontinuities by using a fully automatic hp adaptive finite element method in this paper a fully automatic energy norm based hp adaptive finite element fe strategy applied to a number of relevant waveguide structures is presented the methodology produces exponential convergence rates in terms of the energy norm error of the solution against the problem size number of degrees of freedom extensive numerical results demonstrate the suitability of the hp method for solving different rectangular waveguide discontinuities these results illustrate the flexibility reliability and high accuracy of the method the self adaptive hp fem provides similar sometimes more accurate results than those obtained with semi analytical techniques such as the mode matching method for problems where semi analytical methods can be applied at the same time the hp fem provides the flexibility of modeling more complex waveguide structures and including the effects of dielectrics metallic screws round corners etc which cannot be easily considered when using semi analytical techniques in the second part the work is extended by presenting a fully automatic goal oriented hp adaptive fe in terms of the scattering or s parameters which are widely used in microwave engineering for the characterization of microwave devices

a collocation method for solving some integral equations in distributions a collocation method is presented for the numerical solution of a typical integral equation r h d r x y h y d y f x x d of the class r whose kernels are of positive rational functions of arbitrary selfadjoint elliptic operators defined in the whole space r r and d r r is a bounded domain several numerical examples are given to demonstrate the efficiency and stability of the proposed method

a weak discrete maximum principle for hp fem in this paper we prove a new discrete maximum principle dmp for the one dimensional poisson equation discretized by the hp fem while the dmp for piecewise linear elements is a classical result from the 1970s no extensions to hp fem are available to the present day due to a negative result by hohn and mittelmann from 1981 related to quadratic lagrange elements it was long assumed that higher order finite elements do not satisfy discrete maximum principles in this paper we explain why it is not possible to make a straightforward extension of the classical dmp to the higher order case and we propose stronger assumptions on the right hand side under which an extension is possible

numerical solution of integral equations in a space of distributions abstract the basic integral equation of random fields estimation theory is of the form 1 rh f where r is a linear integral operator which is an isomorphism between the spaces h and h here h d is the sobolev space and h is the dual space of distributions with support in d cl d closure of d 0 and d is the domain of signal processing convergence of the least squares method and a projection method for solving 1 is established the basis functions which describe the correct singularity in the approximate solutions are suggested

a three dimensional self adaptive hp finite element method for the characterization of waveguide discontinuities abstract we propose the use of a highly accurate three dimensional 3d fully automatic hp adaptive finite element method fem for the characterization of rectangular waveguide discontinuities these discontinuities are either the unavoidable result of mechanical electrical transitions or deliberately introduced in order to perform certain electrical functions in modern communication systems the proposed numerical method combines the geometrical flexibility of finite elements with an accuracy that is often superior to that provided by semi analytical methods it supports anisotropic refinements on irregular meshes with hanging nodes and isoparametric elements it makes use of hexahedral elements compatible with high order h curl discretizations the 3d hp adaptive fem is applied for the first time to solve a wide range of 3d waveguide discontinuity problems of microwave communication systems in which exponential convergence of the error is observed

isogeometric finite element data structures based on bezier extraction of t splines we develop finite element data structures for t splines based on bezier extraction generalizing our previous work for nurbs as in traditional finite element analysis the extracted bezier elements are defined in terms of a fixed set of polynomial basis functions the so called bernstein basis the bezier elements may be processed in the same way as in a standard finite element computer program utilizing exactly the same data processing arrays in fact only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same a byproduct of the extraction process is the element extraction operator this operator localizes the topological and global smoothness information to the element level and represents a canonical treatment of t junctions referred to as hanging nodes in finite element analysis and a fundamental feature of t splines a detailed example is presented to illustrate the ideas copyright 2011 john wiley sons ltd

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 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

geometry representation issues associated with p version finite element computations this paper addresses issues related to accurate geometry representation for p version finite elements on curved three dimensional domains specific options to account for domain geometry information during element level computation are identified accuracy requirements on the geometry related approximations to preserve the optimal rate of finite element error convergence for second order elliptic boundary value problems are given an element geometric mapping scheme based on blending the exact shape of the domain boundary is described that can either be used directly during element integrations or used to construct element level geometric approximations of required accuracy smoothness issues of the rational blends on simplex topologies are discussed and a numerical example based on the solution of poisson s equation in three dimensions is presented to illustrate the impact of the rational blends on the optimal rate of finite element error convergence

numerical solution of some integral equations in distributions abstract a class r of integral equations whose kernels are kernels of positive rational functions of elliptic operators do not have in general locally integrable solutions their solutions belong to a distribution space h d where h d is the sobolev space a numerical method is given for computing these solutions results of numerical experiments are reported a typical example of the equation of the class r is the equation 1 1 exp y h y dy 1 1 e h 1 1 1

mesh generation in curvilinear domains using high order elements the ability to construct suitable computational meshes is currently a significant limiting factor in the development of compact high order algorithms in very complex geometries compact high order algorithms such as spectral element or p type finite element techniques offer the potential of high accuracy if the solution is smooth and a well behaved mapping exists between the local sub domains and a standard region in this paper we address the problem of coupling high order algorithms with unstructured mesh generation and cad representation techniques we present three complementary strategies to alleviate the problem of generation of elemental regions with singular elemental mappings the first strategy investigates the influence of an anisometric parametric surface representation on the quality of high order unstructured meshes we show that a straightforward splitting of a standard linear mesh for use in a high order method can lead to the generation of distorted possibly invalid meshes even for simple computational domains such as cubes the second strategy uses hybrid meshes applying prismatic elements near curvilinear boundaries in the third and final strategy we investigate the use of curvature driven surface discretization to incorporate higher order information about the surface into the mesh generation using the computational reconstruction of the arterial bypass graft as an example computational domain we demonstrate how the combination of all three strategies leads to a valid high order discretization of the computational domain copyright 2001 john wiley sons ltd

the hp adaptive fem based on continuous sobolev embeddings isotropic refinements the aim of this paper is to present a new class of smoothness testing strategies in the context of hp adaptive refinements based on continuous sobolev embeddings in addition to deriving a modified form of the 1d smoothness indicators introduced in 26 they will be extended and applied to a higher dimensional framework a few numerical experiments in the context of the hp adaptive fem for a linear elliptic pde will be performed

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

the critical exponents of the two dimensional ising spin glass revisited exact ground state calculations and monte carlo simulations due to an incorrect generation of the bond configuration we obtained incorrect monte carlo results new simulation runs with a corrected and enlarged set of samples has led to the results shown in figures 1 and 2 which replace figures 3 and 4 of the original paper the best data collapse of xl we obtained see figure 1 for and of the binder ratio gl see figure 2 for figure 1 scaling plot of xl for different system sizes and temperatures the value of the correlation length exponent is v 1 5 thus the average value of the correlation length exponent now reads it should be noted that all other results of the paper where independently generated samples were used are not affected by these corrections figure 2 scaling plot of the binder ration gl we obtained the best data collapse for v 1 9 we note that the new estimate is much closer to the results 1 and 2 and is not compatible with the scaling prediction y 1 v

an application of combinatorial optimization to statistical physics and circuit layout design we study the problem of finding ground states of spin glasses with exterior magnetic field and the problem of minimizing the number of vias holes on a printed circuit board or contacts on a chip subject to pin preassignments and layer preferences the former problem comes up in solid state physics and the latter in very large scale integrated vlsi circuit design and in printed circuit board design both problems can be reduced to the max cut problem in graphs based on a partial characterization of the cut polytope we design a cutting plane algorithm and report on computational experience with it our method has been used to solve max cut problems on graphs with up to 1 600 nodes

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

finite element method for solving problems with singular solutions abstract numerical treatment of the elliptic boundary value problem with nonsmooth solution by the finite element method is discussed the nonsmoothness could have its origin in the unsmooth boundary or the differential equation this paper which is a survey of the recent results elaborates among others on the method of auxiliary mapping the partition of unity finite element method and the hp version of fem in three dimensions numerical examples illustrate mathematical results an

regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3 part i countably normed spaces on polyhedral domains this is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in 3 the present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains and it concentrates on exploring the structure of these spaces such as the embeddings of weighted sobolev spaces the relation between weighted sobolev spaces and weighted continuous function spaces and the relations between the weighted sobolev spaces and countably weighted sobolev spaces in cartesian coordinates and in the spherical and cylindrical coordinates these well defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in 3 which are essential for the design of effective computation and the analysis of the h p version of the finite element method for solving elliptic problems in three dimensional nonsmooth domains arising from mechanics and engineering

non degenerated ground states and low degenerated excited states in the antiferromagnetic ising model on triangulations we study the unexpected asymptotic behavior of the degeneracy of the first few energy levels in the antiferromagnetic ising model on triangulations of closed riemann surfaces there are strong mathematical and physical reasons to expect that the number of ground states i e degeneracy of the antiferromagnetic ising model on the triangulations of a fixed closed riemann surface is exponential in the number of vertices in the set of plane triangulations the degeneracy equals the number of perfect matchings of the geometric duals and thus it is exponential by a recent result of chudnovsky and seymour from the physics point of view antiferromagnetic triangulations are geometrically frustrated systems and in such systems exponential degeneracy is predicted we present results that contradict these predictions we prove that for each closed riemann surface s of positive genus there are sequences of triangulations of s with exactly one ground state one possible explanation of this phenomenon is that exponential degeneracy would be found in the excited states with energy close to the ground state energy however as our second result we show the existence of a sequence of triangulations mathcal t n of a closed riemann surface of genus 10 with exactly one ground state such that the degeneracy of each of the 1st 2nd 3rd and 4th excited energy levels belongs to o n o n 2 o n 3 and o n 4 respectively

the p and hp finite element method for problems on thin domains abstract the p and hp versions of the finite element method allow the user to change the polynomial degree to increase accuracy we survey these methods and show how this flexibility can be exploited to counter four difficulties that occur in the approximation of problems over thin domains such as plates beams and shells these difficulties are 1 control of modeling error 2 approximation of corner singularities 3 resolution of boundary layers and 4 control of locking our guidelines enable the efficient resolution of these difficulties when a p hp code is available

a posteriori error estimates for hp finite element solutions of convex optimal control problems in this paper we present a posteriori error analysis for hp finite element approximation of convex optimal control problems we derive a new quasi interpolation operator of clement type and a new quasi interpolation operator of scott zhang type that preserves homogeneous boundary condition the scott zhang type quasi interpolation is suitable for an application in bounding the errors in l 2 norm then hp a posteriori error estimators are obtained for the coupled state and control approximations such estimators can be used to construct reliable adaptive finite elements for the control problems

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

mixed hp finite element approximations on geometric edge and boundary layer meshes in three dimensions in this paper we consider the stokes problem in a three dimensional polyhedral domain discretized with hp finite elements of type qk for the velocity and qk 2 for the pressure defined on hexahedral meshes anisotropically and non quasi uniformly refined towards faces edges and corners the inf sup constant of the discretized problem is independent of arbitrarily large aspect ratios our work generalizes a recent result for two dimensional problems in 10 11

a posteriori error estimates for control problems governed by nonlinear elliptic equations in hp fem abstract in this paper we derive a posteriori error estimates for the hp finite element approximation of the distributed convex optimal control problems governed by nonlinear elliptic equations we obtain hp a posteriori error estimators for both the state and the control approximation these estimates can be used to construct reliable adaptive finite element approximation for the control problems

a numerical study on neumann neumann and feti methods for hp approximations on geometrically refined boundary layer meshes in two dimensions abstract we present extensive numerical tests showing the performance and robustness of certain balancing neumann neumann and one level feti methods for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in two dimensions the numerical results confirm the theoretical results derived in a toselli x vasseur technical report 02 15 seminar fur angewandte mathematik eth zurich september 2002 the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients in addition they only grow polylogarithmically with the polynomial degree as in the case of p approximations on shape regular meshes

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

adaptive hpq finite element methods for the analysis of 3d based models of complex structures part 1 hierarchical modeling and approximations this is a paper out of a series of three papers devoted to model and hpq adaptive finite element methods assigned for modeling and analysis of complex structures in this paper we focus our attention on hierarchical models and hierarchical approximations while the issues of error estimation and adaptivity control will be presented in the next papers of the series in all the papers we consider the elastic structures of complex mechanical description we apply 3d or 3d based mechanical models hierarchical modeling and hierarchical approximations within our finite element formulation in the proposed approach the mechanical model and discretization parameters such as the size h of the element and the longitudinal and transverse approximation orders p and q can vary locally i e they can be different in each finite element the a posteriori error estimation is based on the residual equilibration methods applied to the assessment of the modeling and approximation errors the error controlled adaptive procedures are derived from a four step strategy with possible iterations on h and p steps

factorizations of the s matrix the s matrix of the schrodinger equation regarded as a function on the real axis with values in the group of unitary operators l2 s2 l2 s2 where s2 is the unit sphere in r3 is factorized in two different ways one of these is a standard wiener hopf factorization with respect to the real line the other is the kind of factorization that defines the jost function and which has been found to be a useful tool for the solution of the inverse scattering problem a number of results are given that relate the two factorizations their existence as well as the indices they give rise to some known theorems on the standard factorization lead to new results for the three dimensional inverse scattering problem for the schrodinger equation with a noncentral potential in particular a characterization of admissible s matrices is obtained

on moduli of rings and quadrilaterals algorithms and experiments moduli of rings and quadrilaterals are frequently applied in geometric function theory see e g the handbook by kuhnau handbook of complex analysis geometric function theory vols 1 and 2 north holland amsterdam 2005 yet their exact values are known only in a few special cases previously the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation we present here a new hp fem algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the schwarz christoffel toolbox of driscoll and trefethen we also demonstrate that the hp fem algorithm applies to the case of nonpolygonal boundary and report results with concrete error bounds

fourier series expansion in a non orthogonal system of coordinates for the simulation of 3d dc borehole resistivity measurements abstract we describe a new method to simulate 3d borehole resistivity measurements at zero frequency dc the method combines the use of a fourier series expansion in a non orthogonal system of coordinates with an existing 2d goal oriented higher order self adaptive hp finite element algorithm the new method is suitable for simulating measurements acquired with borehole logging instruments in deviated wells it delivers high accuracy simulations and it enables a considerable reduction of the computational complexity with respect to available 3d simulators since the number of fourier modes basis functions needed to solve practical applications is limited typically below 10 furthermore numerical results indicate that the optimal 2d grid based on the 0th fourier mode also called central fourier mode can be employed to efficiently solve the final 3d problem thereby avoiding the expensive construction of optimal 3d grids specifically for a challenging through casing resistivity application we reduce the computational time from several days using a 3d simulator to just 2 h with the new method while gaining accuracy the new simulation method can be easily extended to different physical phenomena with similar geometries as those arising in the simulation of 3d borehole electrodynamics and sonic acoustics coupled with elasticity measurements in addition the method is especially suited for inversion since we demonstrate that the number of fourier modes needed for the exact representation of the materials is limited to only one the central mode for the case of borehole measurements acquired in deviated wells

3d based hierarchical models and hpq approximations for adaptive finite element method of laplace problems as exemplified by linear dielectricity abstract this paper is devoted to model adaptation and h p q adaptive finite element methods for modeling and analysis of the problems for which the strong formulation corresponds to laplace equation the chosen example of this equation concerns dielectric structures or media of electrostatics the paper addresses hierarchical theories also called hierarchical models and hierarchical approximations in the assessment of the models and approximations our own and existing a priori error estimation results are applied the used assessment procedure can be employed to any other applications of laplace equation in applied sciences the proposed theories understood as mathematical formulations and their numerical approximations are applied to the physical model of linear dielectricity in structures with complex electric description and complex geometry we take advantage of the 3d and 3d based theories hierarchical modeling and hierarchical approximations within h p q finite element formulation in our research the applied theory and discretization parameters i e the element size h the longitudinal approximation order p and the transverse order q differ in each finite element

convergence of the t matrix approach in scattering theory ii the t matrix numerical scheme is widely used in practice convergence of this scheme was not proved a proof of convergence is given in this paper

h p spectral element methods for three dimensional elliptic problems on non smooth domains this paper is devoted to the study of non conforming h p spectral element methods for three dimensional elliptic problems on non smooth domains using parallel computers to overcome the singularities which arise in the neighbourhoods of vertices edges and vertex edges we use local systems of coordinates together with geometric meshes which become finer near corners and edges we provide estimates for the error and the computational complexity of our method and present results of numerical simulations that have been performed to validate the theoretical estimates obtained in 22

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

exact ground states of large two dimensional planar ising spin glasses studying spin glass physics through analyzing their ground state properties has a long history although there exist polynomial time algorithms for the two dimensional planar case where the problem of finding ground states is transformed to a minimum weight perfect matching problem the reachable system sizes have been limited both by the needed cpu time and by memory requirements in this work we present an algorithm for the calculation of exact ground states for two dimensional ising spin glasses with free boundary conditions in at least one direction the algorithmic foundations of the method date back to the work of kasteleyn from the 1960s for computing the complete partition function of the ising model using kasteleyn cities we calculate exact ground states for huge two dimensional planar ising spin glass lattices up to 3000 2 spins within reasonable time according to our knowledge these are the largest sizes currently available kasteleyn cities were recently also used by thomas and middleton in the context of extended ground states on the torus moreover they show that the method can also be used for computing ground states of planar graphs furthermore we point out that the correctness of heuristically computed ground states can easily be verified finally we evaluate the solution quality of heuristic variants of the l bieche et al approach

symmetry properties of scattering amplitudes and applications to inverse problems abstract for the transformations of the potential q q l where l is one of the following transformations rotation scaling or translation the corresponding transformation of the scattering amplitude is given a necessary and sufficient condition on the scattering amplitude for the scatterer to be spherically symmetric is derived as an application of the above results

implicit dynamics in the material point method abstract a time implicit discretization is derived and validated for the material point method mpm the resulting non linear discrete equations are solved using newton s method combined with either the conjugate gradient method or the generalized minimum residual method these newton krylov solvers are implemented in a matrix free fashion for numerical efficiency a description of the algorithms and evaluation of their performance is presented on all test problems if the time step is chosen appropriately the implicit solution technique is more efficient than an explicit method without loss of desired features in the solutions in a dramatic example time steps 10 000 times the explicit step size are possible for the large deformation compression of a cylindrical billet at 1 2 the computational cost

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