a recursive approach to local mesh refinement in two and three dimensions abstract the newest vertex strategy for local refinement in two dimensions is reviewed and discussed a recursive algorithm for local refinement of tetrahedral meshes in three dimensions based on similar bisection strategy is described the recursive approach requires certain restrictions on the initial mesh on the other hand under this condition the refinement process can be kept as local as needed and it can be fully inverted simple data structures and derefinement algorithms are also outlined

compatible strain mixed finite element methods for incompressible nonlinear elasticity abstract we introduce a new family of mixed finite elements for incompressible nonlinear elasticity compatible strain mixed finite element methods csfems based on a hu washizu type functional we write a four field mixed formulation with the displacement the displacement gradient the first piola kirchhoff stress and a pressure like field as the four independent unknowns using the hilbert complexes of nonlinear elasticity which describe the kinematics and the kinetics of motion we identify the solution spaces of the independent unknown fields in particular we define the displacement in h 1 the displacement gradient in h curl the stress in h div and the pressure field in l 2 the test spaces of the mixed formulations are chosen to be the same as the corresponding solution spaces next in a conforming setting we approximate the solution and the test spaces with some piecewise polynomial subspaces of them among these approximation spaces are the tensorial analogues of the nedelec and raviart thomas finite element spaces of vector fields this approach results in compatible strain mixed finite element methods that satisfy both the hadamard compatibility condition and the continuity of traction at the discrete level independently of the refinement level of the mesh by considering several numerical examples we demonstrate that csfems have a good performance for bending problems and for bodies with complex geometries csfems are capable of capturing very large strains and accurately approximating stress and pressure fields using csfems we do not observe any numerical artifacts e g checkerboarding of pressure hourglass instability or locking in our numerical examples moreover csfems provide an efficient framework for modeling heterogeneous solids

a lanczos type method for multiple starting vectors given a square matrix and single right and left starting vectors the classical nonsymmetric lanczos process generates two sequences of bior thogonal basis vectors for the right and left krylov subspaces induced by the given matrix and vectors in this paper we propose a lanczos type algorithm that extends the classical lanczos process for single starting vectors to mul tiple starting vectors given a square matrix and two blocks of right and left starting vectors the algorithm generates two sequences of biorthogonal basis vectors for the right and left block krylov subspaces induced by the given data the algorithm can handle the most general case of right and left start ing blocks of arbitrary sizes while all previously proposed extensions of the lanczos process are restricted to right and left starting blocks of identical sizes other features of our algorithm include a built in deation procedure to detect and delete linearly dependent vectors in the block krylov sequences and the option to employ look ahead to remedy the potential breakdowns that may occur in nonsymmetric lanczos type methods

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

using a dpg method to validate dma experimental calibration of viscoelastic materials abstract a discontinuous petrov galerkin dpg method is used to solve the time harmonic equations of linear viscoelasticity it is based on a broken primal variational formulation which is very similar to the classical primal variational formulation used in galerkin methods but has additional interface variables at the boundaries of the mesh elements both the classical and broken formulations are proved to be well posed in the infinite dimensional setting and the resulting discretization is proved to be stable a full h p convergence analysis is also included and the analysis is verified using computational simulations the method is particularly useful as it carries its own natural arbitrary p a posteriori error estimator which is fundamental for solving problems with localized solution features this proves to be useful when validating calibration models of dynamic mechanical analysis dma experiments indeed different dma experiments of epoxy and silicone resins were successfully validated to within 5 of the quantity of interest using the numerical method

on pade type model order reduction of j hermitian linear dynamical systems a simple yet powerful approach to model order reduction of large scale linear dynamical systems is to employ projection onto block krylov subspaces the transfer functions of the resulting reduced order models of such projection methods can be characterized as pade type approximants of the transfer function of the original large scale system if the original system exhibits certain symmetries then the reduced order models are considerably more accurate than the theory for general systems predicts in this paper the framework of j hermitian linear dynamical systems is used to establish a general result about this higher accuracy in particular it is shown that in the case of j hermitian linear dynamical systems the reduced order transfer functions match twice as many taylor coefficients of the original transfer function as in the general case an application to the sprim algorithm for order reduction of general rcl electrical networks is discussed

geometric computation of curvature driven plane curve evolutions we present a new numerical scheme for planar curve evolution with a normal velocity equal to f kappa where kappa is the curvature and f is a nondecreasing function such that f 0 0 and either x mapsto f x 3 is lipschitz with lipschitz constant less than or equal to 1 or f x x gamma for gamma geq 1 3 the scheme is completely geometrical and avoids some drawbacks of finite difference schemes in particular no special parameterization is needed and the scheme is monotone that is if a curve initially surrounds another one then this remains true during their evolution which guarantees numerical stability we prove consistency and convergence of this scheme in a weak sense finally we display some numerical experiments on synthetic and real data

discrete anisotropic curve shortening flow anisotropic curve shortening flow is a geometric evolution of a curve and is equivalent to the gradient flow of anisotropic interface energy we develop a numerical scheme for this nonlinear and degenerate problem which is based on the fact that the evolution problem can be written formally as a linear partial differential equation on the interface itself the scheme requires the solution of a tridiagonal complex linear system in each time step we prove optimal error estimates in adequate norms for the semidiscrete scheme and provide numerical test computations the scheme can also be applied to crystalline energies

the dpg star method abstract this article introduces the dpg star from now on denoted dpg finite element method it is a method that is in some sense dual to the discontinuous petrov galerkin dpg method the dpg methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem in the same vein the dpg methodology is a means to solve an underdetermined discretization these two viewpoints are developed by embedding the same operator equation into two different saddle point problems the analyses of the two problems have many common elements comparison to other methods in the literature round out the newly garnered perspective notably dpg and dpg methods can be seen as generalizations of l l and least squares methods respectively a priori error analysis and a posteriori error control for the dpg method are considered in detail reports of several numerical experiments are provided which demonstrate the essential features of the new method a notable difference between the results from the dpg and dpg analyses is that the convergence rates of the former are limited by the regularity of an extraneous lagrange multiplier variable

manifold evolution with tangential redistribution of points in this paper we propose several techniques for tangential redistribution of points on evolving surfaces this is an important issue in numerical approximation of any lagrangian evolution model since the quality of the mesh has a significant impact on the result of the computation we explain the volume oriented and length oriented tangential redistribution methods in a general setting of an m dimensional manifold evolving in an n dimensional manifold then we apply the proposed techniques to several manifold evolution problems we explain the numerical approximation of the models and present experiments illustrating the performance of our redistribution techniques

unsymmetric multi level hanging nodes and anisotropic polynomial degrees in h1 conforming higher order finite element methods abstract the implementation of higher order finite element schemes that can handle multi level hanging nodes is known to be a difficult task in fact most of the available literature on hanging nodes in finite element schemes restricts to one level hanging nodes resulting from symmetric bisections the intent of this paper is to provide all data structures and algorithms that are necessary for an implementation of a h 1 conforming higher order finite element method the corresponding finite element spaces are defined via tensor products of hierarchic as well as nodal polynomials for quadrilateral and hexahedron based meshes with unsymmetric multi level hanging nodes and arbitrary anisotropic polynomial degree distributions where special care is given to possible orientation problems the meshes may even result from non matching refinements given these data structures and algorithms an extension for serendipity spaces is described in detail along with some other techniques to improve computational efficiency numerical results from an implementation based on these data structures and algorithms serve as a validation and show the broad possibilities that highly flexible higher order finite element schemes have to offer to the best of our knowledge this paper offers the most comprehensive numerical results so far for various three dimensional benchmark problems using in particular finite element spaces defined for hierarchical as well as nodal polynomials on meshes with unsymmetric refinement ratios as well as multi level hanging nodes most notably the numerical results indicate that unsymmetric refinements are indeed favorable over symmetric refinements with respect to convergence rates however the actual optimal refinement ratio for a given problem seems to depend on the type and magnitude of singularities to be resolved as well as on the chosen full tensor product or serendipity finite element spaces in addition to these numerical results we find that systems of equations defined via finite element spaces using the nodal lagrange polynomials with gauss lobatto quadrature points as support points yield drastically improved condition numbers

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

convergence of a crystalline algorithm for the motion of a closed convex curve by a power of curvature v k alpha a finite difference scheme is constructed for the evolution of a plane curve driven by a power of curvature v k alpha alpha 0 where v and k are the normal velocity and the curvature respectively here we assume that the curve is closed strictly convex and immersed in the plane r2 this curve evolution is discretized using a crystalline approximation which is a kind of finite difference scheme for the nonlinear evolution equation we prove convergence of this scheme and show the rate of convergence moreover some numerical examples are presented

two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence abstract the purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal potential variables one for triangles and the other one for quadrilateral elements to be used in discrete versions of the mixed finite element method for elliptic problems they can be interpreted as enriched versions of b d f m k 1 spaces based on triangles and of r t k spaces for quadrilateral elements the new flux approximations are incremented with properly chosen internal shape functions with vanishing normal components on the edges of degree k 2 and matching primal functions of degree k 1 higher than the border fluxes which are kept of degree k in all these cases the divergence of the flux space coincide with the primal approximation space on the master element producing stable simulations using static condensation the global condensed system to be solved in the enriched cases has same dimension and structure of the original ones which is proportional to the space dimension of the border fluxes for each element geometry measuring the errors with l 2 norms the enriched space configurations give higher convergence rate of order k 2 for the primal variable while keeping the order k 1 for the flux for affine meshes the divergence errors have the same improved accuracy rate as for the error in the primal variable for quadrilateral non affine meshes for instance trapezoidal elements the divergence error has order k 1 one unit more than the order k occurring for r t k spaces on this kind of deformed meshes this fact also holds for a b f k elements but for them the potential order of accuracy does not improve keeping order k 1

hierarchical high order finite element bases for h div spaces based on curved meshes for two dimensional regions or manifolds the mixed finite element formulation for elliptic problems is characterized by simultaneous calculations of the potential primal variable and of the flux field dual variable this work focuses on new h div conforming finite element spaces which are suitable for flux approximations based on curved meshes of a planar region or a manifold domain embedded in r 3 the adopted methodology for the construction of h div bases consists in using hierarchical h 1 conforming scalar bases multiplied by vector fields that are properly constructed on the master element and mapped to the geometrical elements by the piola transformation followed by a normalization procedure they are classified as being of edge or internal type the normal component of an edge function coincides on the corresponding edge with the associated scalar shape function and vanishes over the other edges and the normal components of an internal shape function vanishes on all element edges these properties are fundamental for the global assembly of h div conforming functions locally defined by these vectorial shape functions for applications to the mixed formulation the configuration of the approximation spaces is such that the divergence of the dual space and the primal approximation space coincides results of verification numerical tests are presented for curved triangular and quadrilateral partitions on circular cylindrical and spherical regions demonstrating stable convergence with optimal convergence rates coinciding for primal and dual variables innovative two dimensional hdiv approximation spaces for curved quadrilateral and triangular elements hdiv bases properly constructed on the master element and mapped to the geometrical elements by the piola transformation followed by a normalization procedure stable convergence with optimal convergence rates coinciding for primal and dual variables study of number of condensed equations as a function of the polynomial order of the shape functions

fully automatic hp adaptivity in three dimensions the paper presents a three dimensional implementation of the fully automatic hp adaptive mesh refinement strategy proposed in l demkowicz w rachowicz ph devloo a fully automatic hp adaptivity j sci comput 17 1 3 2002 127 155 the methodology is based on a minimization of the projection based interpolation error with a fixed number of degrees of freedom a coarse grid is first uniformly refined in both h and p to yield a corresponding fine grid a two grid solver is used to solve the problem on the fine grid or iterate towards the fine grid solution the fine grid solution possibly partially converged serves then as a reference solution to be interpolated on the original coarse mesh and the next optimal coarse grid to be determined the difference between the coarse and fine grid solutions provides an excellent error estimate for the coarse grid error the procedure is repeated with the coarse and fine meshes generated during the adaptive process simultaneously until a prescribed error tolerance for the coarse grid error is met the solution obtained on the last fine grid is considered to be the final product delivered by the adaptive procedure

discrete least squares finite element methods abstract a finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators 1 the differential operator defining the spatial boundary value problem and 2 a riesz map on the test space the resulting linear system is overdetermined two different approaches for solving the system are suggested although others are discussed 1 solving the associated normal equation with linear solvers for symmetric positive definite systems e g cholesky factorization and 2 solving the overdetermined system with orthogonalization algorithms e g qr factorization the finite element assembly algorithm for each of these approaches is described in detail the normal equation approach is usually faster for direct solvers and requires less storage the second approach reduces the condition number of the system by a power of two and is less sensitive to round off error the rectangular stiffness matrix of second approach is demonstrated to have condition number o h 1 for a variety of formulations of poisson s equation the stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least squares finite element methods and it is proved that the two are identical in some cases an example with poisson s equation indicates that the solutions of these two different linear systems can be nearly indistinguishable if round off error is not an issue and rapidly converge to each other the orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems experiments with poisson s equation in single precision arithmetic as well as the linear acoustics problem near resonance in double precision arithmetic verify this conclusion the methodology described here is an outgrowth of the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2015 the strength of dpg is highlighted throughout however the majority of theory presented is more general extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments

an implicitly restarted krylov subspace method for real symmetric skew symmetric eigenproblems abstract a new implicitly restarted krylov subspace method for real symmetric skew symmetric generalized eigenvalue problems is presented the new method improves and generalizes the shira method of mehrmann and watkins 2001 37 to the case where the skew symmetric matrix is singular it computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point several applications from control theory are presented and the properties of the new method are illustrated by benchmark examples

hp fem for a stabilized three field formulation of the biharmonic problem abstract in this paper a three field formulation for the biharmonic equation is considered that consists of the potential u the flux and a lagrange multiplier the use of stabilization techniques circumvents the discrete inf sup condition and thus enables the application of arbitrary conforming finite element spaces for the three fields a priori error estimates as well as reliable and efficient a posteriori error estimates are derived the proof of the latter is based on an implicit never to be computed h 2 reconstruction of the discrete potential several numerical experiments underline the theoretical results and show the existence of discretization spaces for which the a posteriori error estimate is p robust with an efficiency index tending towards the optimal value of one the numerical experiments indicate that the proposed method is applicable even if the theoretically required minimal regularity of the lagrange multiplier for the a priori error estimate is not fulfilled in particular h and h p adaptivity recover optimal algebraic and exponential converge rates respectively

krylov subspace methods for reduced order modeling in circuit simulation the simulation of electronic circuits involves the numerical solution of very large scale sparse in general nonlinear systems of differential algebraic equations often the size of these systems can be reduced considerably by replacing the equations corresponding to linear subcircuits by approximate models of much smaller state space dimension in this paper we describe the use of krylov subspace methods for generating such reduced order models of linear subcircuits particular emphasis is on reduced order modeling techniques that preserve the passivity of linear rlc subcircuits

the dpg methodology applied to different variational formulations of linear elasticity abstract the flexibility of the dpg methodology is exposed by solving the linear elasticity equations under different variational formulations including some with non symmetric functional settings different infinite dimensional trial and test spaces the family of formulations presented is proved to be mutually ill or well posed when using traditional energy spaces on the whole domain moreover they are shown to remain well posed when using broken energy spaces and interface variables four variational formulations are solved in 3d using the dpg methodology numerical evidence is given for both smooth and singular solutions and the expected convergence rates are observed

static condensation partial orthogonalization of basis functions and ilu preconditioning in the hp fem static condensation of internal degrees of freedom partial orthogonalization of basis functions and ilu preconditioning are techniques used to facilitate the solution of discrete problems obtained in the hp fem this paper shows that for symmetric linear not necessarily positive definite problems under mild technical assumptions these three techniques are completely equivalent in fact the same matrices can be obtained by the same arithmetic operations the study can be extended to nonsymmetric problems naturally

three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy abstract there are different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal pressure variables to be used in discrete versions of the mixed finite element method for elliptic problems arising in fluid simulations three cases shall be studied and compared for discretized three dimensional formulations based on tetrahedral hexahedral and prismatic meshes the principle guiding the constructions is the property that the divergence of the dual space and the primal approximation space should coincide while keeping the same order of accuracy for the flux variable and varying the accuracy order of the primal variable there is the classic case of b d m k spaces based on tetrahedral meshes and polynomials of total degree k for the dual variable and k 1 for the primal variable showing stable simulations with optimal convergence rates of orders k 1 and k respectively another case is related to r t k and b d f m k 1 spaces for hexahedral and tetrahedral meshes respectively but holding for prismatic elements as well it gives identical approximation order k 1 for both primal and dual variables an improvement in accuracy obtained by increasing the degree of primal functions to k and by enriching the dual space with some properly chosen internal shape functions of degree k 1 while keeping degree k for the border fluxes a new type of approximation is proposed by further incrementing the order of some internal flux functions to k 2 and matching primal functions to k 1 higher than the border fluxes of degree k thus higher convergence rate of order k 2 is obtained for the primal variable using static condensation the global condensed system to be solved in all the cases has same dimension and structure which is proportional to the space dimension of the border fluxes for each element geometry illustrating results comparing the three different space configurations are presented for simulations based on hierarchical high order shape functions for h div conforming spaces which are specially constructed for affine tetrahedral hexahedral and prismatic meshes expected convergence rates are obtained for the flux pressure and divergence variables

a computational method for the coupled solution of reaction diffusion equations on evolving domains and manifolds in this paper we devise a moving mesh finite element method for the approximate solution of coupled bulk surface reaction diffusion equations on an evolving two dimensional domain fundamental to the success of the method is the robust generation of bulk and surface meshes for this purpose we use a novel moving mesh partial differential equation mmpde approach the developed method is applied to model problems with known analytical solutions these experiments indicate second order spatial and temporal accuracy coupled bulk surface problems occur frequently in many areas in particular in the modelling of eukaryotic cell migration and chemotaxis we apply the method to a model of the two way interaction of a migrating cell in a chemotactic field where the bulk region corresponds to the extracellular region and the surface to the cell membrane

the treatment of nonhomogeneous dirichlet boundary conditions by thep version of the finite element method the paper addresses the problem of the implementation of nonhomogeneous essential dirichlet type boundary conditions in thep version of the finite element method

on hamiltonian rational transfer functions abstract a sequel to a previous paper fuhrmann 1983 this paper contains a detailed study of rational transfer functions having hamiltonian symmetry a detailed realization procedure is described that reflects this symmetry as a corollary the williamson canonical form for hamiltonian maps in symplectic spaces is derived

high order composite finite element exact sequences based on tetrahedral hexahedral prismatic pyramidal partitions abstract the combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces this paper presents high order exact sequences of finite element approximations in h 1 h curl h div and l 2 based on such kind of three dimensional mesh configurations the approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra the traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid in order to maintain conformity with shared neighboring hexahedron or prism two classes of composite exact sequences are constructed one using classic nedelec spaces of first kind and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces projection based interpolants satisfying the commuting diagram property are presented in a general form for each type of element the interpolants are expressed as the sum of linearly independent contributions associated with vertices edges faces and volume according to the kind of traces appropriate to the space under consideration furthermore we study applications to the mixed formulation of darcy s problems based on compatible pairs of approximations in h div l 2 for such tetrahedral hexahedral prismatic pyramidal meshes an error analysis is outlined showing same optimal orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions enhanced accuracy for potential and flux divergence variables is obtained when enriched space configurations are applied the predicted convergence orders are verified for some test problems

anisotropic hierarchic solid finite elements for the simulation of passive active arterial wall models abstract a 3d anisotropic hierarchic solid finite element formulation is provided for the passive and active mechanical response of arteries the artery is modeled as an anisotropic nearly incompressible hyperelastic tube consisting of two layers that correspond to the media and the adventitia these layers are considered as a fiber reinforced material consisting of two collagen fiber families that are symmetrically disposed and helically oriented around the tube s axis the numerical analysis is based on a 3d anisotropic hierarchic solid finite element formulation including the possibility of varying the polynomial degree in all local directions as well as for all displacement components for the purpose of verification analytical solutions are provided for different benchmarks focusing on the aspects of compressible and incompressible 3d stretch plane strain and plane stress pure shear as well as a mono layer anisotropic incompressible circular cylindrical artery of holzapfel gasser ogden material under inflation and extension the resulting solutions including stress and strain distribution through the arterial wall are plotted to point out the passive response and different activation levels in addition the effect of various compressibility levels including nearly incompressibility is studied by numerical examples the robustness and accuracy of the proposed method is demonstrated by comparing the results of different ansatz spaces

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

fully discrete hp finite elements fast quadrature a fully discrete hp finite element method fem is presented it combines the features of the standard hp fem conforming galerkin formulation variable order quadrature schemes geometric meshes static condensation and of the spectral element method special shape functions and spectral quadrature techniques the speed up relative to standard hp elements is analyzed in detail both theoretically and computationally

arbitrary level hanging nodes for adaptive hp fem approximations in 3d in this paper we discuss constrained approximation with arbitrary level hanging nodes in adaptive higher order finite element methods hp fem for three dimensional problems this technique enables using highly irregular meshes and it greatly simplifies the design of adaptive algorithms as it prevents refinements from propagating recursively through the finite element mesh the technique makes it possible to design efficient adaptive algorithms for purely hexahedral meshes we present a detailed mathematical description of the method and illustrate it with numerical examples

on convergence of krylov subspace approximations of time invariant self adjoint dynamical systems abstract we extend the rational krylov subspace algorithm from the computation of the action of the matrix exponential to the solution of stable dynamical systems a d dt u t b t u t 0 0 a d dt i 0 m a i d dt si i where m n a i a i r n n s 0 and u t b t r n b t 0 0 not assuming that evolution of b t is described by a low dimensional subspace of r n we show that the reduced equation is stable and derive an a priori error bound via rational approximation of the exponential on the boundary of the nonlinear numerical range of a we also describe a simple and easily computable external bound of this numerical range the obtained results are applied to the infinite order problem arising in the solution of the dispersive maxwell s system

parametric approximation of isotropic and anisotropic elastic flow for closed and open curves deckelnick and dziuk math comput 78 266 645 671 2009 proved a stability bound for a continuous in time semidiscrete parametric finite element approximation of the elastic flow of closed curves in mathbb r d d geq2 we extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in barrett et al j comput phys 222 1 441 462 2007 siam j sci comput 31 1 225 253 2008 ima j numer anal 30 1 4 60 2010 in particular an equidistribution mesh property for this new approximation we obtain also a stability bound for a continuous in time semidiscrete scheme apart from the isotropic situation we also consider the case of an anisotropic elastic energy in addition to the evolution of closed curves we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions

a fictitious domain method for the simulation of thermoelastic deformations in nc milling processes summary this paper presents a higher order finite element approach for the simulation of heat diffusion and thermoelastic deformations in nc milling processes the inherent continuous material removal in the process of the simulation is taken into account via continuous removal dependent refinements of a paraxial hexahedron base mesh covering a given workpiece these refinements rely on isotropic bisections of these hexahedrons along with subdivisions of the latter into tetrahedrons and pyramids in correspondence to a milling surface triangulation obtained from the application of the marching cubes algorithm the resulting mesh is employed for an element wise defined characteristic function for the milling dependent workpiece within that paraxial hexahedron base mesh using this characteristic function a higher order fictitious domain method is used to compute the heat diffusion and thermoelastic deformations where the corresponding ansatz spaces are defined for some hexahedron based refinement of the base mesh numerical experiments compared to real physical experiments exhibit the applicability of the proposed approach to predict deviations of the milled workpiece from its designed shape due to thermoelastic deformations in the process this article is protected by copyright all rights reserved

sparsity optimized high order finite element functions for h div on simplices this paper deals with conforming high order finite element discretizations of the vector valued function space h div in 2 and 3 dimensions a new set of hierarchic basis functions on simplices with the following two main properties is introduced when working with an affine simplicial triangulation first the divergence of the basis functions is l 2 orthogonal and secondly the l 2 inner product of the interior basis functions is sparse with respect to the polynomial degree p the construction relies on a tensor product based construction with properly weighted jacobi polynomials as well as an explicit splitting of the higher order basis functions into solenoidal and non solenoidal ones the basis is suited for fast assembling strategies the general proof of the sparsity result is done by the assistance of computer algebra software tools several numerical experiments document the proved sparsity patterns and practically achieved condition numbers for the parameter dependent div div problem even on curved elements a tremendous improvement in condition numbers is observed the precomputed mass and stiffness matrix entries in general form are available online

second order nedelec curl conforming prismatic element for computational electromagnetics a systematic approach to obtaining mixed order curl conforming basis functions for a triangular prism is presented focus is made on the second order case space of functions for the prism is given basis functions are obtained as the dual basis with respect to suitably discretized nedelec degrees of freedom functionals acting on elements of the space thus the linear independence of the basis functions is assured while the belonging of the basis to the a priori given space of functions is guaranteed different strategies for the finite element assembly of the basis are discussed numerical results showing the verification procedure of the correctness of the implemented basis functions are given numerical results about sensibility of the condition number of the basis obtained concerning the quality of the elements of the mesh are also shown comparison with other representative sets of basis functions for prisms is included

bernstein bezier finite elements on tetrahedral hexahedral pyramidal partitions abstract a construction for high order continuous finite elements on partitions consisting of tetrahedra hexahedra and pyramids based on polynomial bernstein bezier shape functions is presented along with algorithms that allow the computation of the system matrices in optimal complexity o 1 per entry

hierarchic hp edge element families for maxwell s equations on hybrid quadrilateral triangular meshes we construct and study a set of hierarchic basis functions for the galerkin discretisation of the space h curl 169 suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non uniform order polynomial approximation we investigate the conditioning and dispersive behaviour of the elements in addition numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time harmonic maxwell s equations that have singularities

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

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions

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

high order polygonal discontinuous petrov galerkin polydpg methods using ultraweak formulations abstract this work represents the first endeavor in using ultraweak formulations to implement high order polygonal finite element methods via the discontinuous petrov galerkin dpg methodology ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space while most of the trial space can be chosen as copies of l 2 discretizations that have no need to be continuous across adjacent elements additionally the test spaces are broken along the mesh interfaces this allows one to construct conforming polygonal finite element methods termed here as polydpg methods by defining most spaces by restriction of a bounding triangle or box to the polygonal element the only variables that require nontrivial compatibility across elements are the so called interface or skeleton variables which can be defined directly on the element boundaries unlike other high order polygonal methods polydpg methods do not require ad hoc stabilization terms thanks to the crafted stability of the dpg methodology a proof of convergence of the form h p is provided and corroborated through several illustrative numerical examples these include polygonal meshes with n sided convex elements and with highly distorted concave elements as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid since polydpg methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes this work is also accompanied by an open source polydpg software supporting polygonal and conventional elements

about the conditioning of matrices in the p version of the finite element method for second order elliptic problems abstract in the framework of adaptive methods bases of hierarchical type are used in the p version of the finite element method we have studied the matrices corresponding to the most used basis introduced by babuska and szabo in the case of d dimensional rectangular elements for the internal nodes we show that the condition number is o p4 d 1 resp o p4d for the stiffness resp mass matrix moreover we show that the usual diagonal preconditioning divides the exponents by two

new shape functions for triangular p fem using integrated jacobi polynomials in this paper the second order boundary value problem x y u f is discretized by the finite element method using piecewise polynomial functions of degree p on a triangular mesh on the reference element we define integrated jacobi polynomials as interior ansatz functions if is a constant function on each triangle and each triangle has straight edges we prove that the element stiffness matrix has not more than nonzero matrix entries an application for preconditioning is given numerical examples show the advantages of the proposed basis

a block qmr algorithm for non hermitian linear systems with multiple right hand sides abstract many applications require the solution of multiple linear systems that have the same coefficient matrix but differ in their right hand sides instead of applying an iterative method to each of these systems individually it is often more efficient to employ a block version of the method that generates iterates for all the systems simultaneously in this paper we propose a block version of freund and nachtigal s quasi minimal residual qmr method for the iterative solution of non hermitian linear systems the block qmr method uses a novel lanczos type process for multiple starting vectors which was recently developed by aliaga boley freund and hernandez to compute suitable basis vectors for the underlying block krylov subspaces we describe the basic block qmr method and also give important implementation details in particular we show how to incorporate deflation to drop converged linear systems and to delete linearly and almost linearly dependent vectors in the underlying block krylov sequences numerical results are reported that illustrate typical features of the block qmr method

a stable numerical method for the dynamics of fluidic membranes we develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in navier stokes flow local inextensibility of the membrane is ensured by solving a tangential navier stokes equation taking surface viscosity effects of boussinesq scriven type into account in our approach the bulk and surface degrees of freedom are discretized independently which leads to an unfitted finite element approximation of the underlying free boundary problem bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by dziuk numer math 111 55 80 2008 the obtained numerical scheme can be shown to be stable and to have good mesh properties finally the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions the numerical results demonstrate the robustness of the method and it is observed that the conservation properties are fulfilled to a high precision

bernstein bezier finite elements of arbitrary order and optimal assembly procedures algorithms are presented that enable the element matrices for the standard finite element space consisting of continuous piecewise polynomials of degree n on simplicial elements in mathbb r d to be computed in optimal complexity mathcal o n 2d the algorithms i take into account numerical quadrature ii are applicable to nonlinear problems and iii do not rely on precomputed arrays containing values of one dimensional basis functions at quadrature points although these can be used if desired the elements are based on bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

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

orientation embedded high order shape functions for the exact sequence elements of all shapes a unified construction of high order shape functions is given for all four classical energy spaces h1 h curl h div and l2 and for elements of all shapes segment quadrilateral triangle hexahedron tetrahedron triangular prism and pyramid the discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element the shape functions are conforming hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape the polynomial order is allowed to differ for each separate topological entity vertex edge face or interior in the mesh so the shape functions can be used to implement local p adaptive finite element methods each topological entity may have its own orientation and the shape functions can have that orientation embedded by a simple permutation of arguments

an adaptive dpg method for high frequency time harmonic wave propagation problems abstract the discontinuous petrov galerkin dpg method for high frequency wave propagation problems is discussed the dpg method with its attractive uniform mesh and wavenumber independent pre asymptotic stability property allows for a fully automatic adaptive h p algorithm that can be initiated from very coarse meshes moreover dpg always delivers a hermitian positive definite system suggesting the use of the conjugate gradient algorithm for its solution we present a new iterative solution scheme which capitalizes on these attractive properties of dpg this novel solver is integrated within the adaptive procedure by constructing a two grid like preconditioner for the conjugate gradient method that exploits information from previous meshes the construction of our preconditioner is discussed and its efficacy is illustrated with an example of a 2d acoustics problem our results show that the proposed iterative algorithm converges at a rate independent of the mesh and the wavenumber

analysis of non conforming dpg methods on polyhedral meshes using fractional sobolev norms abstract the work is concerned with two problems a analysis of a discontinuous petrov galerkin dpg method set up in fractional energy spaces b use of the results to investigate a non conforming version of the dpg method for general polyhedral meshes we use the ultraweak variational formulation for the model laplace equation the theoretical estimates are supported with 3d numerical experiments