discontinuous finite elements for a hyperbolic problem with singular coefficient a convergence theory for one dimensional spherical transport a theory of convergence is presented for the discontinuous galerkin finite element method of solving the nonscattering spherically symmetric boltzmann transport equation using piecewise constant test and trial functions results are then extended to higher order polynomial spaces comparisons of numerical properties were presented in earlier work

the hough transform versus the upwrite this paper compares the hough transform and the upwrite for the detection of lines circles and ellipses both ideal and noisy images are tested the upwrite is found to be more robust for images containing perturbation noise for ideal images and images with speckle noise the results are found to depend on the complexity of the object being detected with more complex objects favoring the upwrite

adaptive numerical analysis in primal elastoplasticity with hardening the quasi static viscoplastic resp elastoplastic evolution problem with isotropic or kinematic hardening is considered with emphasis on optimal convergence and adapted mesh refining in the spatial discretization within one time step of an implicit time discretization the finite element method leads to a minimisation problem for non smooth convex functions on discrete subspaces for piecewise constant resp affine ansatz functions the stress resp displacement approximations are experimentally and theoretically shown to converge linearly an a posteriori error estimate then justifies an automatic adaptive mesh refining algorithm numerical examples support the superiority of the adapted mesh

a variational formulation of constitutive models and updates in non linear finite viscoelasticity the purpose of this article is to present a general framework for constitutive viscoelastic models in finite strain regime the approach is qualified as variational since the constitutive updates obey a minimum principle within each load increment the set of internal variables is strain based and employs according to the specific model chosen a multiplicative decomposition of strain into elastic and viscous components the present approach shares the same technical procedures used for analogous models of plasticity or viscoplasticity such as the solution of a minimization problem to identify inelastic updates and the use of exponential mapping for time integration however instead of using the classical decomposition of inelastic strains into amplitude and direction we take advantage of a spectral decomposition that provides additional facilities to accommodate into simple analytical expressions a wide set of specific models moreover appropriate choices of the constitutive potentials allow the reproduction of other formulations in the literature the final part of the paper presents a set of numerical examples in order to explore the characteristics of the formulation as well as its applicability to usual large scale fem analyses copyright 2005 john wiley sons ltd

an analysis of some high accuracy finite element methods for hyperbolic problems some high accuracy finite element methods for hyperbolic problems are studied in this paper it is proven that over a finite element mesh the convergence order of linear finite element solutions for both linear and nonlinear equations can be higher than one even if the exact solutions are discontinuous the theoretical tools for the convergence analysis are some superclose error estimates that are also developed in this paper for nonsmooth solutions

consistent elastoplastic cohesive zone model at finite deformations variational formulation cohesive zone models based on classical interface type formulations at finite deformations are subjected to fundamental physical principles such as thermodynamical consistency balance equations and material frame indifference however these restraints are often ignored and in that respect such formulations are inconsistent by way of contrast a consistent cohesive zone framework suitable for the analysis of localized elastoplastic deformations which only depends on the displacement jump was recently advocated in ottosen et al 2015 a certain subclass of this consistent framework is analyzed here further extended and finally an efficient numerical implementation is proposed conceptually the considered cohesive zone model is a fiber like model where the fiber direction is defined by the direction of the displacement discontinuity a novel unloading model is advocated where the key idea is to assign a vanishing bending stiffness to the fibers and they therefore buckle when compressive stresses are initiated following ideas known from wrinkling in membranes it is shown that the resulting framework can be rewritten into a variationally consistent format such that all unknowns follow jointly from minimizing a time dependent potential whose discretization leads to an efficient implementation in terms of an efficient variational constitutive update the physical properties of the final constitutive framework are analyzed by means of numerical examples this analysis shows that although the framework is based on elastoplasticity it predicts for the l shaped structure investigated a mechanical response similar to that of damage theory even during unloading

superconvergence and a posteriori error estimates of the dg method for scalar hyperbolic problems on cartesian grids in this paper we analyze the discontinuous galerkin dg finite element method for the steady two dimensional transport reaction equation on cartesian grids we prove the l2 stability and optimal l2 error estimates for the dg scheme we identify a special numerical flux for which the l2 norm of the solution is of order p 1 when tensor product polynomials of degree at most p are used we further prove superconvergence towards a particular projection of the directional derivative the order of superconvergence is proved to be p 1 2 we also provide a very simple derivative recovery formula which is o h p 1 superconvergent approximation to the directional derivative moreover we establish an o h 2 p 1 global superconvergence for the solution flux at the outflow boundary of the domain these results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element finally we prove that the proposed a posteriori dg error estimates converge to the true errors in the l2 norm at o h p 1 rate and that the global effectivity index converges to unity at o h rate our results are valid without the flow condition restrictions we perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal

constructively well posed approximation methods with unity inf sup and continuity constants for partial differential equations starting from the generalized lax milgram theorem and from the fact that the approximation error is minimized when the continuity and inf sup constants are unity we develop a theory that provably delivers well posed approximation methods with unity continuity and inf sup constants for numerical solution of linear partial differential equations we demonstrate our single framework theory on scalar hyperbolic equations to constructively derive two different hp finite element methods the first one coincides with a least squares discontinuous galerkin method and the other appears to be new both methods are proven to be trivially well posed with optimal hpconvergence rates the numerical results show that our new discontinuous finite element method namely a discontinuous petrov galerkin method is more accurate has optimal convergence rate and does not seem to have unphysical diffusion compared to the upwind discontinuous galerkin method

an adaptive discontinuous finite element method for the transport equation abstract in this paper we introduce a discontinuous finite element method in our approach it is possible to combine the advantages of finite element and finite difference methods the main ingredients are numerical flux approximation and local orthogonal basis functions the scheme is defined on arbitrary triangulations and two different error indicators are derived especially the second one is closely connected to our approach and able to handle arbitrary varying flow directions numerical resuls are given for boundary value problems in two dimensions they demonstrate the performance of the scheme combined with the two error indicators

crosswind smear and pointwise errors in streamline diffusion finite element methods for a model convection dominated singularly perturbed convection diffusion prob lem it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion the ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5 4 under local smoothness assumptions 1 introduction the streamline diffusion method is a finite element method for convection dominated convection diffusion problems which combines formal high accuracy with decent stability properties the method was introduced in the case of stationary problems by hughes and brooks 7 cf raithby and torrance 14 and wahlbin 15 for earlier thoughts in this direction the mathematical analysis of the method was started in johnson and navert 8 and continued with extensions to e g time dependent problems in navert 12 johnson navert and pitkaranta 9 and johnson and saranen 10 in these papers local error estimates in l2 of order o h k 1 2 in regions of smoothness with piecewise polynomial finite elements of degree k were derived together with estimates stating as a typical example that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0 h1 2 essentially the purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0 h3 4 the improvement from 0 h1 2 to 0 h3 4 is obtained by adding a small amount 0 h3 2 of artificial crosswind diffusion to the method in the piecewise linear case k 1 this does not destroy the known o h3 2 accuracy in l2 in smooth regions using our first result we then obtain our second main result localized pointwise error estimates of order 0 h5 4 in regions of smoothness the previously known best pointwise error estimate in the piecewise linear situation is 0 h1 2 another consequence is a global l estimate of order 0 h 2 in the presence of typical crosswind and downwind singularities we shall consider the model problem of finding u u x y such that

superconvergence of discontinuous galerkin and local discontinuous galerkin methods eigen structure analysis based on fourier approach various superconvergence properties of discontinuous galerkin dg and local dg ldg methods for linear hyperbolic and parabolic equations have been investigated in the past due to these superconvergence properties dg and ldg methods have been known to provide good wave resolution properties especially for long time integrations zhong and shu 2011 26 in this paper under the assumption of uniform mesh and via fourier approach we observe that the error of the dg or ldg solution can be decomposed into three parts 1 dissipation and dispersion errors of the physically relevant eigenvalue this part of error will grow linearly in time and is of order 2 k 1 for dg method and 2 k 2 for ldg method 2 projection error there exists a special projection of the exact solution such that the numerical solution is much closer to this special projection than the exact solution itself this part of error will not grow in time 3 the dissipation of non physically relevant eigenvectors this part of error will be damped exponentially fast with respect to the spatial mesh size x along this line we analyze the error for a fully discrete runge kutta rk dg scheme a collection of numerical examples for linear equations are presented to verify our observations above we also provide numerical examples based on non uniform mesh nonlinear burgers equation and high dimensional maxwell equations to explore superconvergence properties of dg methods in a more general setting

a note on the convergence of the discontinuous galerkin method for a scalar hyperbolic equation a numerical example is presented which shows that the known l 2 error estimate left u u h right leqq ch k 1 2 left u right k 1 for the discontinuous galerkin method cannot be improved within the class of quasi uniform meshes not even for smooth exact solutions u

a unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty we discuss stabilized galerkin approximations in a new framework widening the scope from the usual dichotomy of the discontinuous galerkin method on the one hand and petrov galerkin methods such as the supg method on the other the idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation thus circumventing the need of a petrov galerkin type choice of spaces this is made possible by adding a higher order penalty term giving l2 control of the jumps in the gradients between adjacent elements we consider convection diffusion reaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces the standard h1 conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides the crouzeix raviart element moreover we show how the formulation extends to discontinuous galerkin interior penalty methods in a natural way by domain decomposition using nitsche s method

explicit finite element methods for symmetric hyperbolic equations a family of explicit space time finite element methods for the initial boundary value problem for linear symmetric hyperbolic systems of equations is described and analyzed the method generalizes the discontinuous galerkin method and as is typical for this method obtains error estimates of order o h n 1 2 for approximations by polynomials of degree le n

positivity preserving discontinuous galerkin schemes for linear vlasov boltzmann transport equations we develop a high order positivity preserving discontinuous galerkin dg scheme for linear vlasov boltzmann transport equations vlasov bte under the action of quadratically confined electrostatic potentials the solutions of such btes are positive probability distribution functions and it is very challenging to have a mass conservative high order accurate scheme that preserves positivity of the numerical solutions in high dimensions our work extends the maximum principle satisfying scheme for scalar conservation laws in a recent work by x zhang and c w shu to include the linear boltzmann collision term the dg schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy a discussion of the standard semi discrete dg schemes for the bte are included as a foundation for the stability and error estimates for this new scheme numerical results of the relaxation models are provided to validate the method

pointwise error estimates for a streamline diffusion finite element scheme pointwise error estimates for a streamline diffusion scheme for solving a model convection dominated singularly perturbed convection diffusion problem are given these estimates improve pointwise error estimates obtained by johnson et al 5

an optimal order error estimate for the discontinuous galerkin method in this paper a new approach is developed for analyzing the discontinuous galerkin method for hyperbolic equations for a model problem in r2 the method is shown to converge at a rate 0 hn l when applied with nth degree polynomial approxi mations over a semiuniform triangulation assuming sufficient regularity in the solution

edge curvature and convexity based ellipse detection method in this paper we propose a novel ellipse detection method for real images the proposed method uses the information of edge curvature and their convexity in relation to other edge contours as clues for identifying edge contours that can be grouped together a search region is computed for every edge contour that contains other edge contours eligible for grouping with the current edge contour a two dimensional hough transform is performed in an intermediate step in which we use a new relationship score for ranking the edge contours in a group instead of the conventional histogram count the score is found to be more selective and thus more efficient in addition we use three novel saliency criteria that are non heuristic and consider various aspects for quantifying the goodness of the detected elliptic hypotheses and finally selecting good elliptic hypotheses the thresholds for selection of elliptic hypotheses are determined by the detected hypotheses themselves such that the selection is free from human intervention the method requires a few seconds in most cases so it is suitable for practical applications the performance of the proposed ellipse detection method has been tested on a dataset containing 1200 synthetic images and the caltech 256 dataset containing real images in both cases the results show that the proposed ellipse detection method performs far better than existing methods and is close to the ideal results with precision recall and f measure all very close to 1 further the method is robust to the increase in the complexity of the images such as overlapping ellipses occluded ellipses while the performance of the contemporary methods deteriorates significantly

optimal error estimates of the direct discontinuous galerkin method for convection diffusion equations abstract in this paper we present the optimal l2 error estimate ofo hk 1 for polynomial elements of degree k of the semidiscrete direct discontinuous galerkin method for convection diffusion equations the main technical difficulty lies in the control of the inter element jump terms which arise because of the convection and the discontinuous nature of numerical solutions the main idea is to use some global projections satisfying interface conditions dictated by the choice of numerical fluxes so that trouble terms at the cell interfaces are eliminated or controlled in multi dimensional case the orders of k 1 hinge on a superconvergence estimate when tensor product polynomials of degree k are used on cartesian grids a collection of projection errors in both oneand multi dimensional cases is established

error estimates for discontinuous galerkin approximations of implicit parabolic equations we analyze the classical discontinuous galerkin method for implicit parabolic equations symmetric error estimates for schemes of arbitrary order are presented the ideas developed allow certain assumptions frequently required in previous work to be relaxed for example different discrete spaces are allowed at each time step and the spatial operator is not required to be self adjoint or independent of time error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate these projections are local and enjoy optimal approximation properties when the solution is sufficiently regular

error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation this paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions we introduce what we call the geometric corrector which is a sequence associated with every finite volume mesh in mathbf r nd and every nonvanishing vector mathbf a of mathbf r nd first we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size then an order one error estimate for the finite volume scheme occurs afterwards we prove that this norm is indeed bounded by the mesh size in several cases including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases such as the one with independent refined meshes

the weak galerkin finite element method for the transport reaction equation abstract we present and analyze a weak galerkin finite element method for solving the transport reaction equation in d space dimensions this method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon polyhedra we derive the l 2 error estimate of o h k 1 2 order for the discrete solution when the kth order polynomials are used for k 0 moreover for a special class of meshes we also obtain the optimal error estimate of o h k 1 order in the l 2 norm a derivative recovery formula is presented to approximate the convection directional derivative and the corresponding superconvergence estimate is given numerical examples on compatible and non compatible meshes are provided to show the effectiveness of this weak galerkin method

on the numerical implementation of variational arbitrary lagrangian eulerian vale formulations this paper is concerned with the implementation of variational arbitrary lagrangian eulerian formulations also known as variational r adaption methods these methods seek to minimize the energy function with respect to the finite element mesh over the reference configuration of the body we propose a solution strategy based on a viscous regularization of the configurational forces this procedure eliminates the ill posedness of the problem without changing its solutions i e the minimizers of the regularized problems are also minimizers of the original functional we also develop strategies for optimizing the triangulation or mesh connectivity and for allowing nodes to migrate in and out of the boundary of the domain selected numerical examples demonstrate the robustness of the solution procedures and their ability to produce highly anisotropic mesh refinement in regions of high energy density

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

an adaptive higher order method for solving the radiation transport equation on unstructured grids a new class of solution methods based on a short characteristics approach embedded in the finite element framework for the solution of the radiation transport equation is developed and tested the order of convergence and the error to runtime ratio are determined in numerical tests furthermore the efficiency of the scheme is demonstrated in the case of a challenging problem from the field of solar physics a quantitative comparison with other methods based either on the characteristics or on the finite element approach is carried out this emphasizes the efficiency and accuracy of our method two different adaptation mechanisms are studied grid adaptation and local variation of the order of the scheme

error estimates for the discontinuous galerkin methods for parabolic equations the classical discontinuous galerkin method for a general parabolic equation is analyzed symmetric error estimates for schemes of arbitrary order are presented the ideas developed below relax many assumptions required in previous work for example different discrete spaces may be used at each time step and the spatial operator need not be self adjoint or independent of time our error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate these projections are local and enjoy optimal approximation properties when the solution is sufficiently regular

analysis of a combined cg1 dg2 method for the transport equation in this paper we introduce a reduced discontinuous galerkin method in which the space of continuous piecewise linear functions cg1 is enriched with discontinuous piecewise quadratics dg2 the resultant finite element approximation is continuous at the vertices of the mesh and discontinuous across edges faces we analyze the properties of the cg1 dg2 discretization in the context of a steady linear transport equation the presented a priori error estimate shows that the discontinuous enrichment stabilizes the continuous coarse scale component and delivers optimal convergence rates numerical studies for steady and unsteady convection problems confirm this result

superconvergence and a posteriori error estimates for the ldg method for convection diffusion problems in one space dimension in this paper we investigate the superconvergence properties of the local discontinuous galerkin ldg method applied to transient convection diffusion problems in one space dimension we show that the leading terms of the local discretization errors for the p degree ldg solution and its spatial derivative are proportional to p 1 degree right and left radau polynomials respectively thus the discretization errors for the p degree ldg solution and its spatial derivative are o h p 2 superconvergent at the roots of p 1 degree right and left radau polynomials respectively the superconvergence results are used to construct asymptotically correct a posteriori error estimates these error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element numerical computations suggest that these a posteriori ldg error estimates for the solution and its spatial derivative at a fixed time t converge to the true errors at o h p 3 and o h p 2 rates respectively we also show that the global effectivity indices for the solution and its derivative in the l 2 norm converge to unity at o h 2 and o h rates respectively finally we show that the ldg method combined with the a posteriori error estimation procedure yields both accurate error estimates and o h p 2 superconvergent solutions our proofs are valid for arbitrary regular meshes and for p p polynomials with p 1 and for periodic dirichlet and mixed dirichlet neumann boundary conditions several numerical simulations are performed to validate the theory

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

discontinuous galerkin methods for advection diffusion reaction problems we apply the weighted residual approach recently introduced in f brezzi et al comput methods appl mech engrg 195 2006 pp 3293 3310 to derive discontinuous galerkin formulations for advection diffusion reaction problems we devise the basic ingredients to ensure stability and optimal error estimates in suitable norms and propose two new methods

an explicit a priori estimate for a finite volume approximation of linear advection on non cartesian grids we propose an elementary proof of strong convergence for a finite volume approximation of nonstationary linear advection on arbitrary grids in two space dimensions this proof is elementary in the sense that the basic a priori estimate uses only some discrete integrations by parts and the cauchy schwarz inequality numerical results show that the estimate is probably nonoptimal

a posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems abstract in this paper we consider the a posteriori error analysis of discontinuous galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions we establish several residual based a posteriori error estimators which provide global upper bounds and a local lower bound on the error further for nonsteady problem we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space our a posteriori error analysis is based on the mesh dependent a priori estimates for the first order hyperbolic problems these a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods

tangential cover for thick digital curves the recognition of digital shapes is a deeply studied problem the arithmetical framework initiated by reveilles geometrie discrete calcul en nombres entiers et algorithmique these d etat 1991 provides a powerful theoretical basis as well as many algorithms to deal with digital objects the tangential cover first presented in feschet and tougne optimal time computation of the tangent of a discrete curve application to the curvature in g bertrand m couprie l perroton eds 8th discrete geometry for computer imagery lecture notes in computer science vol 1568 springer berlin 1999 pp 31 40 and feschet canonical representations of discrete curves pattern anal appl 8 1 2 2005 84 94 is a useful tool for representing geometric digital primitives it computes the set of all maximal segments of a digital curve and permits either to obtain minimal length polygonalization or asymptotic convergence of tangents estimations nevertheless the arithmetical approach does not tolerate the introduction of irregularities which are however inherent to the acquisition of digital shapes the present paper is an extension of faure and feschet tangential cover for thick digital curves in d coeurjolly i sivignon l tougne f dupont eds dgci 2008 lecture notes in computer science vol 4992 springer berlin 2008 pp 358 369 in which we propose a new definition for a class of the so called thick digital curves that applies well to a large class of digital object boundaries we then propose an extension of the tangential cover to thick digital curves and provide an algorithm with an o nlogn time complexity where n denotes the number of points of specific subparts of the thick digital curve in order to keep up with this low complexity some critical points must be taken into account we describe all required implementation details in this paper

the weak galerkin finite element method for the symmetric hyperbolic systems abstract in this paper we present and analyze a weak galerkin finite element wg method for solving the symmetric hyperbolic systems this method is highly flexible by allowing the use of discontinuous finite elements on element and its boundary independently of each other by introducing special weak derivative we construct a stable weak galerkin scheme and derive the optimal l 2 error estimate of o h k 1 2 order for the discrete solution when the k order polynomials are used for k 0 as application we discuss this wg method for solving the singularly perturbed convection diffusion reaction equation and derive an e uniform error estimate of order k 1 2 numerical examples are provided to show the effectiveness of the proposed wg method

a vertex centered dual discontinuous galerkin method this note introduces a new version of the discontinuous galerkin method for discretizing first order hyperbolic partial differential equations the method uses piecewise polynomials that are continuous on a macroelement surrounding the nodes in the unstructured mesh but discontinuous between the macroelements at lowest order the method reduces to a vertex centered finite volume method with control volumes based on a dual mesh and the method can be implemented using an edge based data structure the method provides therefore a strategy to extend existing vertex centered finite volume codes to higher order using the discontinuous galerkin method preliminary tests on a model linear hyperbolic equation in two dimensional indicate a favorable qualitative behavior for nonsmooth solutions and optimal convergence rates for smooth solutions

error estimates to smooth solutions of runge kutta discontinuous galerkin methods for scalar conservation laws in this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for runge kutta discontinuous galerkin rkdg methods where the time discretization is the second order explicit total variation diminishing tvd runge kutta method error estimates for the mathbb p 1 piecewise linear elements are obtained under the usual cfl condition tau leq gamma h for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions where h and tau are the maximum element lengths and time steps respectively and the positive constant gamma is independent of h and tau however error estimates for higher order mathbb p k k geq 2 elements need a more restrictive time step tau leq gamma h 4 3 we remark that this stronger condition is indeed necessary as the method is linearly unstable under the usual cfl condition tau leq gamma h for the mathbb p k elements of degree k geq 2 error estimates of o h k 1 2 tau 2 are obtained for general monotone numerical fluxes and optimal error estimates of o h k 1 tau 2 are obtained for upwind numerical fluxes

curve fitting approach for tangent angle and curvature measurements abstract the measures of tangent angle and curvature of a digital curve play an important role in image analysis such as corner detection 1d shape representation and shape signature in the generalized hough transform instead of using the discrete measurement approach the least squares method is employed to fit known digital points to two cubic polynomial functions one with y f x that minimizes the sum of the vertical distances and the other with x g y that minimizes the sum of the horizontal distances from the known points to the approximated curve the tangent angle and curvature are therefore directly computable from the first and second order derivatives of the continuous functions hybrid procedures are also proposed to select the better curve from f and g for accurate evaluation of tangent angle and curvature experimented results on both analytic curves and real world images are included

variational h adaption in finite deformation elasticity and plasticity we ropose a variational h adaption strategy in which the evolution of the mesh is driven directly by the governing minimum principle this minimum principle is the principle of minimum potential energy in the case of elastostatics and a minimum principle for the incremental static problem of elasto viscoplasticity in particular the mesh is refined locally when the resulting energy or incremental pseudo energy released exceeds a certain threshold value in order to avoid global recomputes we estimate the local energy released by mesh refinement by means of a lower bound obtained by relaxing a local patch of elements this bound can be computed locally which reduces the complexity of the refinement algorithm to o n we also demonstrate how variational h refinement can be combined with variational r refinement to obtain a variational hr refinement algorithm because of the strict variational nature of the h refinement algorithm the resulting meshes are anisotropic and outperform other refinement strategies based on aspect ratio or other purely geometrical measures of mesh quality the versatility and rate of convergence of the resulting approach are illustrated by means of selected numerical examples copyright 2007 john wiley sons ltd

analysis of the implicit upwind finite volume scheme with rough coefficients we study the implicit upwind finite volume scheme for numerically approximating the linear continuity equation in the low regularity diperna lions setting that is we are concerned with advecting velocity fields that are spatially sobolev regular and data that are merely integrable we prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique distributional solution of the continuous model is at hbox least 1 2 the numerical error is estimated in terms of logarithmic kantorovich rubinstein distances and provides thus a bound on the rate of weak convergence

convergence of an explicit finite volume scheme for first order symmetric systems this paper is devoted to the derivation of a o h1 2 error estimate for the classical upwind explicit in time finite volume scheme for linear first order symmetric systems such a result already existed for the corresponding implicit in time finite volume scheme since it can be interpreted as a particular case of the space time discontinuous galerkin method but the technique of proof used in that case does not extend to explicit schemes the general framework recently developed to analyse the convergence rate of finite volume schemes for non linear scalar conservation laws can not be used either because it is not adapted for systems even linear in this article we propose a new technique which takes advantage of the linearity of the problem the first step consists in controlling the approximation error u u h l2 by an expression of the form 2 where u is the exact solution g is a particular smooth function and h h are some linear forms depending on the approximate solution u h the second step consists in carefully estimating the error terms and by using uniform stability results for the discrete problem and regularity properties of the continuous solution

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

error estimates of the semi discrete local discontinuous galerkin method for nonlinear convection diffusion and kdv equations in this paper we provide l 2 error estimates for the semi discrete local discontinuous galerkin methods for nonlinear convection diffusion equations and kdv equations with smooth solutions the main technical difficulty is the control of the inter element jump terms which arise because of the nonlinearity of the pdes and the discontinuous nature of the numerical method

deb definite error bounded tangent estimator for digital curves we propose a simple and fast method for tangent estimation of digital curves this geometric based method uses a small local region for tangent estimation and has a definite upper bound error for continuous as well as digital conics i e circles ellipses parabolas and hyperbolas explicit expressions of the upper bounds for continuous and digitized curves are derived which can also be applied to nonconic curves our approach is benchmarked against 72 contemporary tangent estimation methods and demonstrates good performance for conic nonconic and noisy curves in addition we demonstrate a good multigrid and isotropic performance and low computational complexity of o 1 and better performance than most methods in terms of maximum and average errors in tangent computation for a large variety of digital curves

a hybridized discontinuous petrov galerkin scheme for scalar conservation laws summary we present a hybridized discontinuous petrov galerkin hdpg method for the numerical solution of steady and time dependent scalar conservation laws the method combines a hybridization technique with a local petrov galerkin approach in which the test functions are computed to maximize the inf sup condition since the petrov galerkin approach does not guarantee a conservative solution we propose to enforce this explicitly by introducing a constraint into the local petrov galerkin problem when the resulting nonlinear system is solved using the newton raphson procedure the solution inside each element can be locally condensed to yield a global linear system involving only the degrees of freedom of the numerical trace this results in a significant reduction in memory storage and computation time for the solution of the matrix system albeit at the cost of solving the local petrov galerkin problems however these local problems are independent of each other and thus perfectly scalable we present several numerical examples to assess the performance of the proposed method the results show that the hdpg method outperforms the hybridizable discontinuous galerkin method for problems involving discontinuities moreover for the test case proposed by peterson the hdpg method provides optimal convergence of order k 1 copyright 2012 john wiley sons ltd

analysis of the cell vertex finite volume method for hyperbolic problems with variable coefficients an analysis of the cell vertex finite volume method is presented for a scalar linear hyperbolic equation with variable coefficients the cell vertex approximation is shown to be second order convergent in the l2 norm and first order convergent in the h1 norm provided the exact solution belongs to h3 and the characteristic curves of the differential equation are transversal to mesh lines on the closure of the domain of integration when characteristics are parallel to mesh lines on a subset of positive measure the scheme is only first order convergent in the l2 norm however if transversality fails on part of the outflow boundary only then the cell vertex scheme is second order convergent in a weighted l2 norm

adaptive least squares finite element methods for linear transport equations based on an h div flux reformulation abstract in this paper we study the least squares finite element methods lsfem for the linear hyperbolic transport equations the linear transport equation naturally allows discontinuous solutions and discontinuous inflow conditions while the normal component of the flux across the mesh faces needs to be continuous traditional lsfems using continuous finite element approximations will introduce unnecessary extra error for discontinuous solutions and boundary conditions in order to separate the continuity requirements a new flux variable is introduced with this reformulation the continuities of the flux and the solution can be handled separately in natural h div l 2 conforming finite element spaces several variants of the methods are developed to handle the inflow boundary condition strongly or weakly with the reformulation the new lsfems can handle discontinuous solutions and boundary conditions much better than the traditional lsfems with continuous polynomial approximations with least squares functionals as a posteriori error estimators the adaptive methods can naturally identify error sources including singularity and non matching discontinuity for discontinuity aligned mesh no extra error is introduced if an r t 0 p 0 pair is used to approximate the flux and the solution the new adaptive lsfems can approximate discontinuous solutions with almost no overshooting even when the mesh is not aligned with discontinuity existence and uniqueness of the solutions and a priori and a posteriori error estimates are established for the proposed methods extensive numerical tests are performed to show the effectiveness of the methods developed in the paper

a novel variational algorithmic formulation for wrinkling at finite strains based on energy minimization application to mesh adaption this paper is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains in contrast to previously published numerical implementations the advocated method is fully variational more precisely the parameters describing wrinkles or slacks together with the unknown deformation mapping are computed jointly by minimizing the potential energy of the considered mechanical system furthermore the wrinkling criteria are naturally included within the presented variational framework the proposed method allows to employ three dimensional constitutive models without any additional modification i e a projection in plane stress space is not required analogously to the wrinkling parameters the non vanishing out of plane component of the strain tensor results conveniently from relaxing the respective helmholtz energy of the membrane the proposed framework is very general and does not rely on any assumption regarding the symmetry of the material i e arbitrary anisotropic hyperelastic models can be consistently taken into account the advantages associated with such a variational method are manifold for instance it opens up the possibility of applying standard optimization algorithms to the numerical implementation this is especially important for highly non linear or singular problems such as wrinkling on the other hand minimization principles provide a suitable basis for a posteriori error estimation and thus for adaptive finite element formulations as a prototype a variational error indicator leading to an efficient h adaption for wrinkling is briefly discussed the performance of the wrinkling approach is demonstrated by selected finite element analyses

optimal convergence of the original dg method for the transport reaction equation on special meshes we show that the approximation given by the original discontinuous galerkin method for the transport reaction equation in d space dimensions is optimal provided the meshes are suitably chosen the l 2 norm of the error is of order k 1 when the method uses polynomials of degree k these meshes are not necessarily conforming and do not satisfy any uniformity condition they are required only to be made of simplexes each of which has a unique outflow face we also find a new element by element postprocessing of the derivative in the direction of the flow which superconverges with order k 1

taming the cfl number for discontinuous galerkin methods on structured meshes the upwind discontinuous galerkin method is an attractive method for solving time dependent hyperbolic conservation laws it is possible to use high order explicit time stepping methods and high order spatial approximations without incurring heavy numerical linear algebra overheads however the courant friedrichs lewy cfl condition for these methods depends on the polynomial order used and there is a somewhat excessive cost for using very high order spatial approximation we discuss the impact of a covolume mesh based filter on the cfl number for these methods and present an algorithm which has a cfl number independent of the spatial order of approximation we present computational results for the advection equation and the wave equation on one dimensional meshes using up to tenth order in space and time

how accurate is the streamline diffusion finite element method we investigate the optimal accuracy of the streamline diffusion finite element method applied to convection dominated problems for linear bilinear elements the theoretical order of convergence given in the literature is either o h 3 2 for quasi uniform meshes or o h 2 for some uniform meshes the determination of the optimal order in general was an open problem by studying a special type of meshes it is shown that the streamline diffusion method may actually converge with any order within this range depending on the characterization of the meshes

a variational formulation for finite deformation wrinkling analysis of inelastic membranes this paper is concerned with a novel fully variational formulation for finite deformation analysis of inelastic membranes with wrinkling in contrast to conventional approaches every aspect of the physical problem derives from minimization of suitable energy functionals a variational formulation of finite strain plasticity theory which leads to a minimization problem for the constitutive updates serves as the starting point for the derivations in order to take into account the kinematics induced by wrinkles and slacks a relaxed version of the finite strain functional is postulated in effect the local incremental stress strain relations are established via differentiation of the relaxed energy functional with respect to the strains hence the presented formulation is fully analogous to that of hyperelasticity with the sole exception that the aforementioned functional depends on history variables and accordingly it is path dependent the advantages associated with the developed variational method are manifold from a practical point of view the possibility of applying standard optimization algorithms to solve the minimization problem describing inelastic membranes is remunerative from a mathematical point of view on the other hand the energy of the system induces some sort of natural metric representing an essential requirement for error estimation and thus for adaptive finite element methods the presented derivation of the model allows to consider possible material symmetries in the elastic as well as plastic response of the material as a prototype a von mises type model is implemented the efficiency and performance of the resulting algorithm are demonstrated by means of numerical examples

non convex potentials and microstructures in finite strain plasticity a mathematical model for a finite strain elastoplastic evolution problem is proposed in which one time step of an implicit time discretization leads to generally non convex minimization problems the elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations the results for a single slip system and von mises plasticity illustrate that finite strain elastoplasticity generates reduced problems with non quasiconvex energy densities and so allows for non attainment of energy minimizers and microstructures

superconvergence of discontinuous galerkin methods for hyperbolic systems in this paper the discontinuous galerkin method for the positive and symmetric linear hyperbolic systems is constructed and analyzed by using bilinear finite elements on a rectangular domain and an o h 2 order superconvergence error estimate is established under the conditions of almost uniform partition and the h 3 regularity for the exact solutions the convergence analysis is based on some superclose estimates derived in this paper finally as an application the numerical treatment of maxwell equation is discussed and computational results are presented

on the order of convergence of the discontinuous galerkin method for hyperbolic equations the basic error estimate for the discontinuous galerkin method for hyperbolic equations indicates an o h n convergence rate for nth degree polynomial approximation over a triangular mesh of size h however the optimal o h 1 rate is frequently seen in practice here we extend the class of meshes for which sharpness of the o h estimate can be demonstrated using as an example a problem with a nonaligned mesh in which all triangle sides are bounded away from the characteristic direction the key to realizing h n convergence is a mesh which to the extent possible directs the error to lower frequency modes which are approximated not damped as h 0

superconvergence of discontinuous galerkin methods for scalar nonlinear conservation laws in one space dimension in this paper an analysis of the superconvergence property of the semidiscrete discontinuous galerkin dg method applied to one dimensional time dependent nonlinear scalar conservation laws is carried out we prove that the error between the dg solution and a particular projection of the exact solution achieves left k frac32 right th order superconvergence when upwind fluxes are used the results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree k k ge 1 under the condition that f u possesses a uniform positive lower bound numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions indicating that the restriction on f u is artificial

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

comparison and improvement of tangent estimators on digital curves many contour based applications rely on the estimation of the geometry of the shape such as pattern recognition or classification methods this paper proposes a comprehensive evaluation on the problem of tangent estimators on digital curves the methods taken into account use different paradigms approximation and digital geometry in the former paradigm methods based on polynomial fitting smoothing and filtering are reviewed in the latter case of digital geometry we consider two methods that mainly rely on digital straight line recognition j o lachaud a vialard f de vieilleville fast accurate and convergent tangent estimation on digital contours image and vision computing 25 10 2007 1572 1587 and optimization b kerautret j o lachaud robust estimation of curvature along digital contours with global optimization in proceedings of discrete geometry for computer imagery lyon france lecture notes in computer science vol 4992 springer berlin 2008 the comparison takes into account objective criteria such as multi grid convergence average error maximum error isotropy and length estimation experiments underline that adaptive methods based on digital straight line recognition often propose a good trade off between time and precision and that if precision is to be sought non adaptive methods can be easily transformed into adaptive methods to get more accurate estimations

curvature and tangential deflection of discrete arcs a theory based on the commutator of scatter matrix pairs and its application to vertex detection in planar shape data this paper introduces a new theory for the tangential deflection and curvature of plane discrete curves our theory applies to discrete data in either rectangular boundary coordinate or chain coded formats its rationale is drawn from the statistical and geometric properties associated with the eigenvalue eigenvector structure of sample covariance matrices specifically we prove that the nonzero entry of the commutator of a piar of scatter matrices constructed from discrete arcs is related to the angle between their eigenspaces and further we show that this entry is in certain limiting cases also proportional to the analytical curvature of the plane curve from which the discrete data are drawn these results lend a sound theoretical basis to the notions of discrete curvature and tangential deflection and moreover they provide a means for computationally efficient implementation of algorithms which use these ideas in various image processing contexts as a concrete example we develop the commutator vertex detection cvd algorithm which identifies the location of vertices in shape data based on excessive cummulative tangential deflection and we compare its performance to several well established corner detectors that utilize the alternative strategy of finding approximate curvature extrema

convexity conditions for strain dependent energy functions for membranes the forms that the convexity polyconvexity and rank one convexity inequalities take when the strain energy is required to be a function of the strain g are studied it is shown in particular that w g must be an increasing function of g in the sense that w g w g if g g is non negative definite relatively simple sufficient conditions in terms of g alone are given necessary and sufficient conditions in terms of g alone are found to be rather complex

tension field theory a general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations the principal contribution is a partial differential equation describing a geometrical property of tension trajectories this is one of a system of two equations which describes the state of stress independently of the deformation this system is strongly elliptic at any stable solution whereas the deformation is described by a system of parabolic type controllable solutions i e those states that can be maintained in any isotropic elastic material by application of edge tractions and lateral pressure alone are obtained the general axisymmetric problem is solved implicitly and the theory is applied to the solution of two representative examples existing small strain theories are shown to correspond to a singular limit of the general theory at which the underlying system changes from elliptic to parabolic type

the variational formulation of viscoplastic constitutive updates we present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations non newtonian viscosity rate sensitivity and arbitrary flow and hardening rules the distinguishing characteristic of the proposed constitutive updates is that by construction the corresponding incremental stress strain relations derive from a pseudo elastic strain energy density this in turn confers the incremental boundary value problem a variational structure in particular the incremental deformation mapping follows from a minimum principle this minimum principle may conveniently be taken as a basis for error estimation and mesh adaption the accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely deforming mises solid and ductile single crystals the ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy

convergence of a fully discrete scheme for two dimensional neutron transport we prove an error estimate for a fully discrete method for the numerical solution of a two dimensional model problem in neutron transport theory based on using the discrete ordinates method for the

an upwind like discontinuous galerkin method for hyperbolic systems we investigate an upwind like dg method for solving first order hyperbolic problems written as the friedrichs systems under certain condition this dg scheme may be semi explicit such that the discrete equations can be solved layer by layer we give the stability analysis and error estimate of order k 1 2 in the dg norm in particular for some hyperbolic systems we show that the convergence rate is of order k 1 in the l 2 norm if the q k elements are used on rectangular meshes finally we provide some numerical experiments to illustrate the theoretical analysis

nonuniform time step runge kutta discontinuous galerkin method for computational aeroacoustics with many superior features runge kutta discontinuous galerkin method rkdg which adopts discontinuous galerkin method dg for space discretization and runge kutta method rk for time integration has been an attractive alternative to the finite difference based high order computational aeroacoustics caa approaches however when it comes to complex physical problems especially the ones involving irregular geometries the time step size of an explicit rk scheme is limited by the smallest grid size in the computational domain demanding a high computational cost for obtaining time accurate numerical solutions in caa for computational efficiency high order rk method with nonuniform time step sizes on nonuniform meshes is developed in this paper in order to ensure correct communication of solutions on the interfaces of grids with different time step sizes the values at intermediate stages of the runge kutta time integration on the elements neighboring such interfaces are coupled with minimal dissipation and dispersion errors based upon the general form of an explicit p stage rk scheme a linear coupling procedure is proposed with details on the coefficient matrices and execution steps at common time levels and intermediate time levels applications of the coupling procedures to runge kutta schemes frequently used in simulation of fluid flow and acoustics are given including the third order tvd scheme and low storage low dissipation and low dispersion lddrk schemes in addition an analysis on the stability of coupling procedures on a nonuniform grid is carried out for validation numerical experiments on one dimensional and two dimensional problems are presented to illustrate the stability and accuracy of proposed nonuniform time step rkdg scheme as well as the computational benefits it brings application to a one dimensional nonlinear problem is also investigated

optimal convergence of the original dg method on special meshes for variable transport velocity we prove optimal convergence rates for the approximation provided by the original discontinuous galerkin method for the transport reaction problem this is achieved in any dimension on meshes related in a suitable way to the possibly variable velocity carrying out the transport thus if the method uses polynomials of degree k the l 2 norm of the error is of order k 1 moreover we also show that by means of an element by element postprocessing a new approximate flux can be obtained which superconverges with order k 1

implicit runge kutta methods and discontinuous galerkin discretizations for linear maxwell s equations in this paper we consider implicit runge kutta methods for the time integration of linear maxwell s equations we first present error bounds for the abstract cauchy problem which respect the unboundedness of the differential operators using energy techniques the error bounds hold for algebraically stable and coercive methods such as gauss and radau collocation methods the results for the abstract evolution equation are then combined with a discontinuous galerkin discretization in space using upwind fluxes for the case that permeability and permittivity are piecewise constant functions we show error bounds for the full discretization where the constants do not deteriorate if the spatial mesh width tends to zero

optimal error estimate and superconvergence of the dg method for first order hyperbolic problems we consider the original discontinuous galerkin method for the first order hyperbolic problems in d dimensional space we show that when the method uses polynomials of degree k the l 2 error estimate is of order k 1 provided the triangulation is made of rectangular elements satisfying certain conditions further we show the o h 2 k 1 order superconvergence for the error on average on some suitably chosen subdomains including the whole domain and their outflow faces moreover we also establish a derivative recovery formula for the approximation of the convection directional derivative which is superconvergent with order k 1

scale based description and recognition of planar curves and two dimensional shapes the problem of finding a description at varying levels of detail for planar curves and matching two such descriptions is posed and solved in this paper a number of necessary criteria are imposed on any candidate solution method path based gaussian smoothing techniques are applied to the curve to find zeros of curvature at varying levels of detail the result is the generalized scale space image of a planar curve which is invariant under rotation uniform scaling and translation of the curve these properties make the scale space image suitable for matching the matching algorithm is a modification of the uniform cost algorithm and finds the lowest cost match of contours in the scale space images it is argued that this is preferable to matching in a so called stable scale of the curve because no such scale may exist for a given curve this technique is applied to register a landsat satellite image of the strait of georgia b c manually corrected for skew to a map containing the shorelines of an overlapping area