minimum cross entropy thresholding abstract the threshold selection problem is solved by minimizing the cross entropy between the image and its segmented version the cross entropy is formulated in a pixel to pixel basis between the two images and a computationally attractive algorithm employing the histogram is developed without making a priori assumptions about the population distribution this method provides an unbiased estimate of a binarized version of the image in an information theoretic sense

a characterization of hybridized mixed methods for second order elliptic problems in this paper we give a new characterization of the approximate solution given by hybridized mixed methods for second order self adjoint elliptic problems we apply this characterization to obtain an explicit formula for the entries of the matrix equation for the lagrange multiplier unknowns resulting from hybridization we also obtain necessary and sufficient conditions under which the multipliers of the raviart thomas and the brezzi douglas marini methods of similar order are identical

locally discontinuous but globally continuous galerkin methods for elliptic problems abstract we propose and analyze a stabilized hybrid finite element method for elliptic problems consisting of locally discontinuous galerkin problems in the primal variable coupled to a globally continuous problem in the multiplier numerical analysis shows that the proposed formulation preserves the main properties of the associate dg method such as consistency stability boundedness and optimal rates of convergence in the energy norm and in the l 2 norm for adjoint consistent formulations for using an element based data structure it has basically the same complexity and computational cost of classical conforming finite element methods convergence studies confirm the optimal rates of convergence predicted by the numerical analysis presented here with accuracy equivalent or even better than the corresponding dg approximations

a superconvergent ldg hybridizable galerkin method for second order elliptic problems we identify and study an ldg hybridizable galerkin method which is not an ldg method for second order elliptic problems in several space dimensions with remarkable convergence properties unlike all other known discontinuous galerkin methods using polynomials of degree k 0 for both the potential as well as the flux the order of convergence in l 2 of both unknowns is k 1 moreover both the approximate potential as well as its numerical trace superconverge in l 2 like norms to suitably chosen projections of the potential with order k 2 this allows the application of element by element postprocessing of the approximate solution which provides an approximation of the potential converging with order k 2 in l 2 the method can be thought to be in between the hybridized version of the raviart thomas and that of the brezzi douglas marini mixed methods

thresholding based on variance and intensity contrast a new thresholding criterion is formulated for segmenting small objects by exploring the knowledge about intensity contrast it is the weighted sum of within class variance and intensity contrast between the object and background theoretical bounds of the weight are given for the uniformly distributed background and object followed by the procedure to estimate the weight from prior knowledge tests against two real and two synthetic images show that small objects can be extracted successfully irrespective of the complexity of background and difference in class sizes

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

on 2d elliptic discontinuous galerkin methods we discuss the discretization using discontinuous galerkin dg formulation of an elliptic poisson problem two commonly used dg schemes are investigated the original average flux proposed by bassi and rebay j comput phys 1997 131 267 and the local discontinuous galerkin ldg siam j numer anal 1998 35 2440 2463 scheme in this paper we expand on previous expositions discontinuous galerkin methods theory computation and applications springer berlin 2000 135 146 siam j sci comput 2002 24 2 524 547 int j numer meth engng 2003 58 2 1119 1148 by adopting a matrix based notation with a view to highlighting the steps required in a numerical implementation of the dg method through consideration of standard c0 type expansion bases as opposed to elementally orthogonal expansions with the matrix formulation we are able to apply static condensation techniques to improve efficiency of the direct solver when high order expansions are adopted the use of c0 type expansions also permits the direct enforcement of dirichlet boundary conditions through a lifting approach where the ldg flux does not require further stabilization in our construction we also adopt a formulation of the continuous dg fluxes that permits a more general interpretation of their numerical implementation in particular it allows us to determine the conditions under which the ldg method provides a near local stencil finally a study of the conditioning and the size of the null space of the matrix systems resulting from the dg discretization of the elliptic problem is undertaken copyright 2005 john wiley sons ltd

a hybrid mixed method for the compressible navier stokes equations we present a novel discretization method for nonlinear convection diffusion equations and in particular for the compressible navier stokes equations the method is based on a discontinuous galerkin dg discretization for convection terms and a mixed method using h div spaces for the diffusive terms furthermore hybridization is used to reduce the number of globally coupled degrees of freedom for the scalar case a local postprocessing procedure is used to enhance the quality of the approximate solution w h the method reduces to a dg scheme for pure convection and to a mixed method for pure diffusion while for the intermediate case the combined variational formulation requires no additional parameters we formulate and validate our scheme for nonlinear model problems as well as compressible flow problems

a multiscale discontinuous galerkin method for convection diffusion reaction problems we provide a general framework of multiscale discontinuous galerkin methods developed in buffa et al 2006 hughes et al 2006 for general second order partial differential equations we establish stability of the method and prove the error estimates

t tests f tests and otsu s methods for image thresholding otsu s binarization method is one of the most popular image thresholding methods student s t test is one of the most widely used statistical tests to compare two groups this paper aims to stress the equivalence between otsu s binarization method and the search for an optimal threshold that provides the largest absolute student s t statistic it is then naturally demonstrated that the extension of otsu s binarization method to multi level thresholding is equivalent to the search for optimal thresholds that provide the largest f statistic through one way analysis of variance anova furthermore general equivalences between some parametric image thresholding methods and the search for optimal thresholds with the largest likelihood ratio test statistics are briefly discussed

simulation of groundwater age distributions the objective of our work is to examine how to simulate the age of groundwater in such a way that it can be compared to actual measurements we start by showing that computation of kinematic age the one obtained by tracking water along streamlines is ill posed in heterogeneous aquifers this together with its inability to account for mixing processes makes it inadequate for comparison with age measurements which are the result of some averaging of the age distribution in the water sample the type of averaging depends on the measurement procedure therefore we go on to write the equations for the cumulative distribution function of residence time under transient flow conditions this allows us to derive transient equations for the mean age as well as for the higher order moments of its distribution which generalize previous results by others these moments can be used for approximating age measurements which need not be equal to the mean age of the water sample using both a synthetic and a real example we show that mean age is an acceptable estimate of radiometric age measurements in many cases including a second order correction variance of residence time distribution always improves results on the other hand higher order approximations converge slowly for old compared to half life waters to the point that third order approximations often worsen the results

boundary lagrange multipliers in finite element methods error analysis in natural norms a finite element procedure for circumventing the babuska brezzi condition in mixed formulations with lagrange multipliers defined on the boundary is presented residual terms constructed from the euler lagrange equations are added to the classical galerkin formulation in order to attain coercivity in a mesh dependent norm convergence is proven for the primal variable and the multiplier in the natural mesh independent norm of the problem generalizing results of a previous paper

cauchy data requirement of the inverse problem of the mean age equation a classical theoretical inverse problem in groundwater hydraulics is the calibration of a heterogeneous isotropic hydraulic conductivity field using known hydraulic head data at each point in the domain it has been long known that a necessary condition for the uniqueness of the solution to this theoretical inverse problem is the cauchy data i e the hydraulic conductivity values or equivalently flux along a line crossing all streamtubes in the entire domain this result has been cited in identifying the value of boundary flux data in practical inverse problems although in general such data are generally not available in this work we show that groundwater mean age data replace this cauchy data requirement

a posteriori error estimation for discontinuous galerkin solutions of hyperbolic problems abstract we analyze the spatial discretization errors associated with solutions of one dimensional hyperbolic conservation laws by discontinuous galerkin methods dgms in space we show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a radau polynomial of degree p 1 on each element we also prove that the local and global discretization errors are o x2 p 1 and o x2p 1 at the downwind point of each element this strong superconvergence enables us to show that local and global discretization errors converge as o xp 2 at the remaining roots of radau polynomial of degree p 1 on each element convergence of local and global discretization errors to the radau polynomial of degree p 1 also holds for smooth solutions as p these results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth

a galerkin interface stabilisation method for the advection diffusion and incompressible navier stokes equations abstract a stabilised galerkin finite element method for the advection diffusion and incompressible navier stokes equations is presented which inherits features of both discontinuous and continuous galerkin methods the problem of interest is posed element wise with weakly imposed dirichlet boundary conditions the dirichlet boundary condition is supplied by a function on element boundaries which is determined such that weak continuity of the flux is assured the approach allows the natural incorporation of upwinding at element boundaries which is typical of discontinuous galerkin methods while retaining the same number of global degrees of freedom as for a continuous galerkin method the formulation also stabilises mixed incompressible problems which use equal order interpolations for linear elements only minor modifications are required to existing continuous finite element codes and the link to other stabilised methods is elaborated the method is supported by a range of numerical examples which demonstrate stability with minimal numerical dissipation

discontinuous subgrid formulations for transport problems in this paper we develop two discontinuous galerkin formulations within the framework of the two scale subgrid method for solving advection diffusion reaction equations we reformulate using broken spaces the nonlinear subgrid scale nsgs finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh here two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges the amount of subgrid viscosity is scaled by the resolved scale solution at the element level yielding a self adaptive method so that no additional stabilization parameter is required numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous galerkin methods

a projection based error analysis of hdg methods we introduce a new technique for the error analysis of hybridizable discontinuous galerkin hdg methods the technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods this renders the analysis of the projections of the discretization errors simple and concise by showing that these projections of the errors are bounded in terms of the distance between the solution and its projection our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection we illustrate the technique on a specific hdg method applied to a model second order elliptic problem

continuous discontinuous galerkin methods stabilized through transfer functions applied to the incompressible elasticity and to the stokes problem abstract in this work we present a family of new continuous discontinuous mixed formulations with interpolation of equal order for velocity and pressure for the stokes problem and incompressible elasticity the degrees of freedom associated to the discontinuous component are eliminated and an approach with computational effort similar to that of the continuous formulation is obtained an analysis of continuity and weak coercivity infsup condition is performed and it is proved that the formulation is continuous and satisfies the infsup condition on an appropriated norm error estimates derived from theorems and the lemma presented are obtained on a similar norm to the one of the gls method numerical experiments with the cavity problem and a smooth solution problem show the robustness of the new formulation

a multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method proliferation of degrees of freedom has plagued discontinuous galerkin methodology from its inception over 30 years ago this paper develops a new computational formulation that combines the advantages of discontinuous galerkin methods with the data structure of their continuous galerkin counterparts the new method uses local element wise problems to project a continuous finite element space into a given discontinuous space and then applies a discontinuous galerkin formulation the projection leads to parameterization of the discontinuous degrees of freedom by their continuous counterparts and has a variational multiscale interpretation this significantly reduces the computational burden and at the same time little or no degradation of the solution occurs in fact the new method produces improved solutions compared with the traditional discontinuous galerkin method in some situations

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

a hybridizable discontinuous galerkin method for steady state convection diffusion reaction problems in this article we propose a novel discontinuous galerkin method for convection diffusion reaction problems characterized by three main properties the first is that the method is hybridizable this renders it efficiently implementable and competitive with the main existing methods for these problems the second is that when the method uses polynomial approximations of the same degree for both the total flux and the scalar variable optimal convergence properties are obtained for both variables this is in sharp contrast with all other discontinuous methods for this problem the third is that the method exhibits superconvergence properties of the approximation to the scalar variable this allows us to postprocess the approximation in an element by element fashion to obtain another approximation to the scalar variable which converges faster than the original one in this paper we focus on the efficient implementation of the method and on the validation of its computational performance with this aim e

high order discontinuous galerkin methods with lagrange multiplier for hyperbolic systems of conservation laws in this work we present novel high order discontinuous galerkin methods with lagrange multiplier dglm for hyperbolic systems of conservation laws lagrange multipliers are introduced on the inter element boundaries via the concept of weak divergence static condensation on element unknowns considerably reduces globally coupled degrees of freedom resulting in the stiffness equations in the lagrange multipliers only we first establish stability results and provide conditions on the stabilization parameter which plays an important role in resolving discontinuities as well accuracy tests are then performed which shows optimal convergence in the l2 norm extensive numerical results indicate that the dglm has potentials in delivering high order accurate information for various problems in hyperbolic conservation laws numerical examples include inviscid burgers equations shallow water equations subcritical flow and supercritical upstream subcritical downstream flow and 2d circular dam break and compressible euler equations intersection of mach 3 and sods shock tube

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

hdg pod reduced order model of the heat equation abstract we propose a new hybridizable discontinuous galerkin hdg model order reduction technique based on proper orthogonal decomposition pod we consider the heat equation as a test problem and prove error bounds that converge to zero as the number of pod modes increases we present 2d and 3d numerical results to illustrate the convergence analysis

high order dg dglm method for hyperbolic conservation laws abstract a high order discontinuous galerkin method in time combined with discontinuous galerkin method with lagrange multiplier dglm kim 2015 4 in space is proposed to approximate the solution to hyperbolic conservation laws with boundary conditions stability of the approximate solution is proved in a broken l 2 l 2 norm and also in an l l 2 norm error estimates of o h r 1 2 k n q 1 2 with p r e and p q j n elements r q d 1 2 are derived in a broken l 2 l 2 norm where h and k n are the maximum diameters of the elements and the time step of j n respectively j n is the time interval and d is the dimension of the spatial domain an explanation on algorithmic aspects is given

hybridizable discontinuous galerkin projection methods for navier stokes and boussinesq equations schemes for the incompressible navier stokes and boussinesq equations are formulated and derived combining the novel hybridizable discontinuous galerkin hdg method a projection method and implicit explicit runge kutta imex rk time integration schemes we employ an incremental pressure correction and develop the corresponding hdg finite element discretization including consistent edge space fluxes for the velocity predictor and pressure correction we then derive the proper forms of the element local and hdg edge space final corrections for both velocity and pressure including the hdg rotational correction we also find and explain a consistency relation between the hdg stability parameters of the pressure correction and velocity predictor we discuss and illustrate the effects of the time splitting error we then detail how to incorporate the hdg projection method time split within standard imex rk time stepping schemes our high order hdg projection schemes are implemented for arbitrary mixed element unstructured grids with both straight sided and curved meshes in particular we provide a quadrature free integration method for a nodal basis that is consistent with the hdg method to prevent numerical oscillations we develop a selective nodal limiting approach its applications show that it can stabilize high order schemes while retaining high order accuracy in regions where the solution is sufficiently smooth we perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented to complete these studies and to illustrate a range of properties for our new schemes we employ an unsteady tracer advection benchmark a manufactured solution for the steady diffusion and stokes equations and a standard lock exchange boussinesq problem

a phase based hybridizable discontinuous galerkin method for the numerical solution of the helmholtz equation we introduce a new hybridizable discontinuous galerkin hdg method for the numerical solution of the helmholtz equation over a wide range of wave frequencies our approach combines the hdg methodology with geometrical optics in a fashion that allows us to take advantage of the strengths of these two methodologies the phase based hdg method is devised as follows first we enrich the local approximation spaces with precomputed phases which are solutions of the eikonal equation in geometrical optics second we propose a novel scheme that combines the hdg method with ray tracing to compute multivalued solution of the eikonal equation third we utilize the proper orthogonal decomposition to remove redundant modes and obtain locally orthogonal basis functions which are then used to construct the global approximation spaces of the phase based hdg method and fourth we propose an appropriate choice of the stabilization parameter to guarantee stability and accuracy for the proposed method numerical experiments presented show that optimal orders of convergence are achieved that the number of degrees of freedom to achieve a given accuracy is independent of the wave number and that the number of unknowns required to achieve a given accuracy with the proposed method is orders of magnitude smaller than that with the standard finite element method

a weakly compressible hybridizable discontinuous galerkin formulation for fluid structure interaction problems abstract a scheme for the solution of fluid structure interaction fsi problems with weakly compressible flows is proposed in this work a novel hybridizable discontinuous galerkin hdg method is derived for the discretization of the fluid equations while the standard continuous galerkin cg approach is adopted for the structural problem the chosen hdg solver combines robustness of discontinuous galerkin dg approaches in advection dominated flows with higher order accuracy and efficient implementations two coupling strategies are examined in this contribution namely a partitioned dirichlet neumann scheme in the context of hybrid hdg cg discretizations and a monolithic approach based on nitsche s method exploiting the definition of the numerical flux and the trace of the solution to impose the coupling conditions numerical experiments show optimal convergence of the hdg and cg primal and mixed variables and superconvergence of the postprocessed fluid velocity the robustness and the efficiency of the proposed weakly compressible formulation in comparison to a fully incompressible one are also highlighted on a selection of two and three dimensional fsi benchmark problems

an implicit high order hybridizable discontinuous galerkin method for nonlinear convection diffusion equations in this paper we present hybridizable discontinuous galerkin methods for the numerical solution of steady and time dependent nonlinear convection diffusion equations the methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary applying the newton raphson procedure and the hybridization technique we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every newton iteration the high number of globally coupled degrees of freedom in the discontinuous galerkin approximation is therefore significantly reduced we then extend the method to time dependent problems by approximating the time derivative by means of backward difference formulae when the time marching method is p 1 th order accurate and when polynomials of degree p 0 are used to represent the scalar variable each component of the flux and the approximate trace we observe that the approximations for the scalar variable and the flux converge with the optimal order of p 1 in the l 2 norm finally we apply element by element postprocessing schemes to obtain new approximations of the flux and the scalar variable the new approximate flux which has a continuous interelement normal component is shown to converge with order p 1 in the l 2 norm the new approximate scalar variable is shown to converge with order p 2 in the l 2 norm the postprocessing is performed at the element level and is thus much less expensive than the solution procedure for the time dependent case the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required extensive numerical results are provided to demonstrate the performance of the present method

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

a stabilised finite element method for the convection diffusion reaction equation in mixed form abstract this paper is devoted to the approximation of the convection diffusion reaction equation using a mixed first order formulation we propose and analyse a stabilised finite element method that allows equal order interpolations for the primal and dual variables this formulation reminiscent of the galerkin least squares method is proven stable and convergent in addition a numerical assessment of the numerical performance of different stabilised finite element methods for the mixed formulation is carried out and the different methods are compared in terms of accuracy stability and sharpness of the layers for two different classical test problems

optimal threshold selection for tomogram segmentation by projection distance minimization grey value thresholding is a segmentation technique commonly applied to tomographic image reconstructions many procedures have been proposed to optimally select the grey value thresholds based on the tomogram data only e g using the image histogram in this paper a projection distance minimization pdm method is presented that uses the tomographic projection data to determine optimal thresholds these thresholds are computed by minimizing the distance between the forward projection of the segmented image and the measured projection data an important contribution of the current paper is the efficient implementation of the forward projection method which makes the use of the original projection data as a segmentation criterion feasible simulation experiments applied to various phantom images show that our proposed method obtains superior results compared to established histogram based projection data methods

adaptive simulations of two phase flow by discontinuous galerkin methods in this paper we present and compare primal discontinuous galerkin formulations of the two phase flow equations the wetting phase pressure and saturation equations are decoupled and solved sequentially proposed adaptivity in space and time techniques yield accurate and efficient solutions slope limiters valid on non conforming meshes are also presented numerical examples of homogeneous and heterogeneous media are considered

a discontinuous galerkin method with lagrange multipliers for spatially dependent advection diffusion problems abstract a higher accuracy discontinuous galerkin method with lagrange multipliers dglm is presented for the solution of the advection diffusion equation with a spatially varying advection field in the high peclet number regime where the classical polynomial finite element method fem produces spurious oscillations in the solution at practical mesh resolutions the proposed dglm method is based on discontinuous polynomial shape functions that are attached to an element rather than its nodes it overcomes the aforementioned spurious oscillation issue by enriching these functions with approximate free space solutions of homogeneous equations derived from an asymptotic analysis of the governing partial differential equation inspired by prandtl s boundary layer theory these enrichment functions are capable of resolving exponential parabolic and corner boundary layers at relatively coarse mesh resolutions the proposed method enforces a weak continuity of the solution approximation across inter element boundaries using polynomial lagrange multipliers which makes it a hybrid method however unlike other hybrid methods it operates directly on the second order form of the advection diffusion equation and does not require any stabilization its intrinsic performance and its superiority over the higher order polynomial fem are demonstrated for several test problems at peclet numbers ranging from one thousand to one billion

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

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

superconvergent discontinuous galerkin methods for second order elliptic problems we identify discontinuous galerkin methods for second order elliptic problems in several space dimensions having superconvergence properties similar to those of the raviart thomas and the brezzi douglas marini mixed methods these methods use polynomials of degree k 0 for both the potential as well as the flux we show that the approximate flux converges in l2 with the optimal order of k 1 and that the approximate potential and its numerical trace superconverge in l2 like norms to suitably chosen projections of the potential with order k 2 we also apply element by element postprocessing of the approximate solution to obtain new approximations of the flux and the potential the new approximate flux is proven to have normal components continuous across inter element boundaries to converge in l2 with order k 1 and to have a divergence converging in l2 also with order k 1 the new approximate potential is proven to converge with order k 2 in l2 numerical experiments validating these theoretical results are presented

a discontinuous subgrid eddy viscosity method for the time dependent navier stokes equations in this paper we provide an error analysis of a subgrid scale eddy viscosity method using discontinuous polynomial approximations for the numerical solution of the incompressible navier stokes equations optimal continuous in time error estimates of the velocity are derived the analysis is completed with some error estimates for two fully discrete schemes which are first and second order in time respectively

optimum image thresholding via class uncertainty and region homogeneity thresholding is a popular image segmentation method that converts a gray level image into a binary image the selection of optimum thresholds has remained a challenge over decades besides being a segmentation tool on its own often it is also a step in many advanced image segmentation techniques in spaces other than the image space we introduce a thresholding method that accounts for both intensity based class uncertainty a histogram based property and region homogeneity an image morphology based property a scale based formulation is used for region homogeneity computation at any threshold intensity based class uncertainty is computed by fitting a gaussian to the intensity distribution of each of the two regions segmented at that threshold the theory of the optimum thresholding method is based on the postulate that objects manifest themselves with fuzzy boundaries in any digital image acquired by an imaging device the main idea here is to select that threshold at which pixels with high class uncertainty accumulate mostly around object boundaries to achieve this a threshold energy criterion is formulated using class uncertainty and region homogeneity such that at any image location a high energy is created when both class uncertainty and region homogeneity are high or both are low finally the method selects that threshold which corresponds to the minimum overall energy the method has been compared to a maximum segmented image information method superiority of the proposed method was observed both qualitatively on clinical medical images as well as quantitatively on 250 realistic phantom images generated by adding different degrees of blurring noise and background variation to real objects segmented from clinical images

an hdg method for distributed control of convection diffusion pdes abstract we propose a hybridizable discontinuous galerkin hdg method to approximate the solution of a distributed optimal control problem governed by an elliptic linear convection diffusion pde we use degree k polynomials to approximate the state adjoint state their fluxes and the optimal control and we show the approximations converge with order k 1 in the l 2 norm finally we use a simple element by element postprocessing scheme to obtain new superconvergent approximations of the state dual state and the control we show the postprocessed variables converge with order k 2 in the l 2 norm we present 2d and 3d numerical experiments to illustrate our theoretical results

hessian based model reduction for large scale systems with initial condition inputs reduced order models that are able to approximate output quantities of interest of high fidelity computational models over a wide range of input parameters play an important role in making tractable large scale optimal design optimal control and inverse problem applications we consider the problem of determining a reduced model of an initial value problem that spans all important initial conditions and pose the task of determining appropriate training sets for reduced basis construction as a sequence of optimization problems we show that under certain assumptions these optimization problems have an explicit solution in the form of an eigenvalue problem yielding an efficient model reduction algorithm that scales well to systems with states of high dimension furthermore tight upper bounds are given for the error in the outputs of the reduced models the reduction methodology is demonstrated for a large scale contaminant transport problem

the direct inverse problem in aquifers solution of the inverse problem for aquifers requires identification of a distributed parameter system in addition to ill posedness the inverse problem may be underdetermined if recharge is treated as a spatially and temporally variable distribution parameter galerkin s method is applied to a weak formulation of the direct inverse problem leading to a possibly nonlinear weighted least squares problem consideration of generic errors suggests that if the estimate for some derivative of head contains a significant error on one element for all time then significant estimation errors are possible for all nodes and for all reservoir parameters in contrast if some derivative of head is small in some spatial region for all time causing the matrix to be inverted in the least squares algorithm to be nearly singular then significant estimation errors can occur in the corresponding reservoir parameters in that spatial region although it is still possible that other reservoir parameters in that region as well as all reservoir parameters elsewhere may be satisfactorily estimated

a space time hybridizable discontinuous galerkin method for incompressible flows on deforming domains we present the first space time hybridizable discontinuous galerkin hdg finite element method for the incompressible navier stokes and oseen equations major advantages of a space time formulation are its excellent capabilities of dealing with moving and deforming domains and grids and its ability to achieve higher order accurate approximations in both time and space by simply increasing the order of polynomial approximation in the space time elements our formulation is related to the hdg formulation for incompressible flows introduced recently in e g n c nguyen j peraire b cockburn a hybridizable discontinuous galerkin method for stokes flow comput methods appl mech eng 199 2010 582 597 however ours is inspired in typical dg formulations for compressible flows which allow for a more straightforward implementation another difference is the use of polynomials of fixed total degree with space time hexahedral and quadrilateral elements instead of simplicial elements we present numerical experiments in order to assess the quality of the performance of the methods on deforming domains and to experimentally investigate the behavior of the convergence rates of each component of the solution with respect to the polynomial degree of the approximations in both space and time

an analysis of the embedded discontinuous galerkin method for second order elliptic problems the embedded discontinuous galerkin methods are obtained from hybridizable discontinuous galerkin methods by a simple change of the space of the hybrid unknown in this paper we consider embedded methods for second order elliptic problems obtained from hybridizable discontinuous methods by changing the space of the hybrid unknown from discontinuous to continuous functions this change results in a significantly smaller stiffness matrix whose size and sparsity structure coincides with those of the stiffness matrix of the statically condensed continuous galerkin method it is shown that this computational advantage has to be balanced against the fact that the approximate solutions for the scalar variable and its flux lose each a full order of convergence indeed we prove that if polynomials of degree k ge1 are used for the original hybridizable discontinuous galerkin method its approximations to the scalar variable and its flux converge with order k 2 and k 1 respectively whereas those of the corresponding embedded discontinuous galerkin method converge with orders k 1 and k respectively only we also provide numerical results comparing the relative efficiency of the methods

a discontinuous galerkin method with lagrange multiplier for hyperbolic conservation laws with boundary conditions we introduce a discontinuous galerkin method with lagrange multiplier dglm to approximate the solution to the hyperbolic conservation laws with boundary conditions lagrange multipliers are introduced on the edge face of the element via weak divergence wang and ye 2014 the final global system has reduced numbers of unknowns of the standard dg methods numerical fluxes from finite volume difference method are not considered for the time discretization backward euler difference method is used stability of the approximate solution is proved in energy norm discontinuity of the solution is allowed in the error analysis local error estimates of o hr 12 t with pr e elements rd 12 are derived where h and t are the maximum diameter of the elements and time steps respectively and d is the dimension of the spatial domain the high order approximation is obtained under an appropriate condition on the stabilizing parameter it is shown that the method preserves the property of the local mass conservation an explanation on algorithmic aspects is given

stabilization mechanisms in discontinuous galerkin finite element methods in this paper we propose a new general framework for dg methods which allows to uncover a basic mechanism that ensures suitable stability properties of the methods we show that such a mechanism is essentially the same for apparently unrelated stabilizations such as jump penalty upwinding and hughes franca type residual based stabilizations

threshold selection based on fuzzy c partition entropy approach abstract thresholding is an important topic for image processing pattern recognition and computer vision selecting thresholds is a critical issue for many applications the fuzzy set theory has been successfully applied to many areas such as control image processing pattern recognition computer vision medicine social science etc it is generally believed that image processing bears some fuzziness in nature in this paper we use the concept of fuzzy c partition and the maximum fuzzy entropy principle to select threshold values for gray level images we have conducted experiments on many images the experimental results demonstrate that the proposed approach can select the thresholds automatically and effectively and the resulting images can preserve the main features of the components of the original images very well

analysis of an interface stabilized finite element method the advection diffusion reaction equation analysis of an interface stabilized finite element method for the scalar advection diffusion reaction equation is presented the method inherits attractive properties of both continuous and discontinuous galerkin methods namely the same number of global degrees of freedom as a continuous galerkin method on a given mesh and the stability properties of discontinuous galerkin methods for advection dominated problems simulations using the approach in other works demonstrated good stability properties with minimal numerical dissipation and standard convergence rates for the lowest order elements were observed in this work stability of the formulation in the form of an inf sup condition for the hyperbolic limit and coercivity for the elliptic case is proved as is order k 1 2 order convergence for the advection dominated case and order k 1 convergence for the diffusive limit in the l 2 norm the analysis results are supported by a number of numerical experiments

analysis of a multiscale discontinuous galerkin method for convection diffusion problems we study a multiscale discontinuous galerkin method introduced in t j r hughes g scovazzi p bochev and a buffa comput meth appl mech engrg 195 2006 pp 2761 2787 that reduces the computational complexity of the discontinuous galerkin method seemingly without adversely affecting the quality of results for a stabilized variant we are able to obtain the same error estimates for the convection diffusion equation as for the usual discontinuous galerkin method we assess the stability of the unstabilized case numerically and find that the inf sup constant is positive bounded uniformly away from zero and very similar to that for the usual discontinuous galerkin method

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

primal stabilized hybrid and dg finite element methods for the linear elasticity problem abstract primal stabilized hybrid finite element methods for the linear elasticity problem are proposed consisting of locally discontinuous galerkin problems in the primal variable coupled to a global problem in the multiplier which is identified with the trace of the displacement field numerical analysis covering both continuous or discontinuous interpolations of the multiplier shows that the proposed formulation preserves the main properties of the associate dg method such as consistency stability boundedness and optimal rates of convergence in the energy norm and in the l 2 norm for adjoint consistent formulations convergence studies confirm the optimal rates of convergence predicted by the numerical analysis presented here and a local post processing technique is proposed to recover stress approximations with improved rates of convergence in h div norm

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

a discontinuous finite element based domain decomposition method in this work we introduce a somewhat unconventional finite element based nonoverlapping domain decomposition method the convenient space decomposition method csd for parallel computers that can be applied to a wide class of boundary value problems and which can only be implemented efficiently exclusively with primal variables by means of the use of discontinuous elements interesting aspects and properties are pointed out along with proofs for the most relevant results numerical experiments for purely diffusion problems are also presented in order to evaluate the effectiveness of the proposed scheme and to confirm the validity of the related theoretical results

an explicit hybridizable discontinuous galerkin method for the acoustic wave equation abstract we present an explicit hybridizable discontinuous galerkin hdg method for numerically solving the acoustic wave equation the method is fully explicit high order accurate in both space and time and coincides with the classic discontinuous galerkin dg method with upwinding fluxes for a particular choice of its stabilization function this means that it has the same computational complexity as other explicit dg methods however just as its implicit version it provides optimal convergence of order k 1 for all the approximate variables including the gradient of the solution and when the time stepping method is of order k 2 it displays a superconvergence property which allow us by means of local postprocessing to obtain new improved approximations of the scalar field variables at any time levels for which an enhanced accuracy is required in particular the new approximations converge with order k 2 in the l 2 norm for k 1 these properties do not hold for all numerical fluxes indeed our results show that when the hdg numerical flux is replaced by the lax friedrichs flux the above mentioned superconvergence properties are lost although some are recovered when the lax friedrichs flux is used only in the interior of the domain finally we extend the explicit hdg method to treat the wave equation with perfectly matched layers we provide numerical examples to demonstrate the performance of the proposed method

a superconvergent hybridisable discontinuous galerkin method for linear elasticity this is the peer reviewed version of the following article sevilla r giacomini m karkoulias a huerta a a superconvergent hybridisable discontinuous galerkin method for linear elasticity international journal for numerical methods in engineering 12 octubre 2018 vol 116 num 2 p 91 116 which has been published in final form at https doi org 10 1002 nme 5916 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

on the design of discontinuous galerkin methods for elliptic problems based on hybrid formulations abstract the objective of this paper is to present a new framework for the design of discontinuous galerkin dg methods for elliptic problems the idea is to start from a hybrid formulation of the problem involving as unknowns the main field in the interior of the element domains and its fluxes and traces on the element boundaries rather than working with this three field formulation fluxes are modeled using finite difference expressions and then the traces are determined by imposing continuity of fluxes although other strategies could be devised this procedure is applied to four elliptic problems namely the convection diffusion equation in the diffusion dominated regime the stokes problem the darcy problem and the maxwell problem we justify some well known dg methods with some modifications that in fact allow to improve the performance of the original methods particularly when the physical properties are discontinuous

continuous discontinuous galerkin methods applied to elasticity problems abstract a new family of coupled continuous discontinuous galerkin formulations is presented and analyzed in this paper these formulations have some distinguishing properties support to all boundary conditions without differentiating whether the condition is dirichlet or not the continuous part of the formulations can use the discontinuous part to have better accuracy and robustness properties and the discontinuous part has the same stabilized properties of common discontinuous galerkin methods a new promising stability parameter is introduced and its effects analyzed in the numerical experiments some important results related to the stability of the formulations associated with the polynomial degree adopted for the continuous component are also shown

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

a family of mixed finite elements for the elasticity problem a new mixed finite element formulation for the equations of linear elasticity is considered in the formulation the variables approximated are the displacement the unsymmetric stress tensor and the rotation the rotation act as a lagrange multiplier introduced in order to enforce the symmetry of the stress tensor based on this formulation a new family of both two and three dimensional mixed methods is defined optimal error estimates which are valid uniformly with respect to the poisson ratio are derived finally a new postprocessing scheme for improving the displacement is introduced and analyzed

an improved iterative hdg approach for partial differential equations abstract we propose and analyze an iterative high order hybridized discontinuous galerkin ihdg discretization for linear partial differential equations we improve our previous work 45 in several directions 1 the improved ihdg approach converges in a finite number of iterations for the scalar transport equation 2 it is unconditionally convergent for both the linearized shallow water system and the convection diffusion equation 3 it has improved stability and convergence rates 4 we uncover a relationship between the number of iterations and time stepsize solution order meshsize and the equation parameters this allows us to choose the time stepsize such that the number of iterations is approximately independent of the solution order and the meshsize and 5 we provide both strong and weak scalings of the improved ihdg approach up to 16 384 cores a connection between ihdg and time integration methods such as parareal and implicit explicit methods are discussed extensive numerical results for linear and nonlinear pdes are presented to verify the theoretical findings

a high order discontinuous galerkin method for unsteady advection diffusion problems abstract a high order discontinuous galerkin method with lagrange multipliers is presented for the solution of unsteady advection diffusion problems in the high peclet number regime it operates directly on the second order form of the governing equation and does not require any stabilization its spatial basis functions are chosen among the free space solutions of the homogeneous form of the partial differential equation obtained after time discretization it also features lagrange multipliers for enforcing a weak continuity of the approximated solution across the element interface boundaries this leads to a system of differential algebraic equations which are time integrated by an implicit family of schemes the numerical stability of these schemes and the well posedness of the overall discretization method are supported by a theoretical analysis the performance of this method is demonstrated for various high peclet number constant coefficient model flow problems

a limiting approach for dg discretizations on mixed type meshes abstract a limiting procedure of discontinuous galerkin dg discretizations that is suitable for computations of high speed flows with strong shocks around complex geometries using a p adaptive procedure on mixed type quadrilateral triangular meshes is developed a total variation bounded tvb approach is the building block of the limiting procedure the main difference of this work with other recent limiting approaches for dg discretizations that were based on tvb limiting is that a unified approach for limiting on mixed type meshes is obtained this is achieved by applying the tvb limiter on the transformed canonical elements where appropriate estimates of the variation of the approximate solution derivatives are feasible applications for mixed type meshes and p adaptive calculations with the dg method are performed for a number of standard test cases for shock capturing schemes to illustrate the potential of the proposed limiting approach it is demonstrated in the computed results that good resolution of discontinuities and embedded smooth flow features is obtained with quadrilateral triangular and mixed type meshes furthermore the resolution of the present approach on arbitrary type meshes is equally good with the resolution obtained with other limiting approaches of equivalent order obtained on rectangular meshes

symmetric and nonsymmetric discontinuous galerkin methods for reactive transport in porous media for solving reactive transport problems in porous media we analyze three primal discontinuous galerkin dg methods with penalty namely symmetric interior penalty galerkin sipg nonsymmetric interior penalty galerkin nipg and incomplete interior penalty galerkin iipg a cut off operator is introduced in dg to treat general kinetic chemistry error estimates in l2 h1 are established which are optimal in h and nearly optimal in p we develop a parabolic lift technique for sipg which leads to h optimal and nearly p optimal error estimates in the l2 l2 and negative norms numerical results validate these estimates we also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces

a high order hybridizable discontinuous galerkin method for elliptic interface problems we present a high order hybridizable discontinuous galerkin method for solving elliptic interface problems in which the solution and gradient are nonsmooth because of jump conditions across the interface the hybridizable discontinuous galerkin method is endowed with several distinct characteristics first they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries thereby leading to a significant reduction in the global degrees of freedom second they provide for elliptic problems with polygonal interfaces approximations of all the variables that converge with the optimal order of kc1 in the l norm where k denotes the polynomial order of the approximation spaces third they possess some superconvergence properties that allow the use of an inexpensive element by element postprocessing to compute a new approximate solution that converges with order kc2 however for elliptic problems with finite jumps in the solution across the curvilinear interface the approximate solution and gradient do not converge optimally if the elements at the interface are isoparametric the discrepancy between the exact geometry and the approximate triangulation near the curved interfaces results in lower order convergence to recover the optimal convergence for the approximate solution and gradient we propose to use superparametric elements at the interface copyright 2012 john wiley sons ltd

a continuous time inverse operator for groundwater and contaminant transport modeling deterministic case the value of time dependent data in the calibration of numerical models of groundwater flow has been recently highlighted in the literature in this paper the background for a new development in the inverse problem is placed and a new operator for inversion of the spatially discretized diffusion equation is introduced the motivation for the development of an inverse problem solution procedure which uses information in transient data is illustrated with a focused review of recent literature such an approach which implicitly incorporates the continuous time dependency of the flow model is then presented this approach is based upon the new concept of inverting the model after the for example finite element spatial discretization but before the typically finite difference temporal discretization of the model not afterward as has been done in the past the laplace transform is used to replace discretization in time of the system of ordinary differential equations to discretization in the laplace variable of the algebraic system in laplace space the existence of the deterministic solution in the laplace transform space does not require knowledge of the eigenvalues of the diffusion matrix but only the specification of the laplace variables near zero a simple numerical example from the literature is used to generate a set of three simplified test cases upon which the proposed method is tested the implications of the theoretical investigation and the results of the numerical experiments are discussed with a view toward prospects of practical implementation

the compact discontinuous galerkin cdg method for elliptic problems we present a compact discontinuous galerkin cdg method for an elliptic model problem the problem is first cast as a system of first order equations by introducing the gradient of the primal unknown or flux as an additional variable a standard discontinuous galerkin dg method is then applied to the resulting system of equations the numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level thus resulting in a global system that involves only the original unknown variable the proposed method is closely related to the local discontinuous galerkin ldg method b cockburn and c w shu siam j numer anal 35 1998 pp 2440 2463 but unlike the ldg method the sparsity pattern of the cdg method involves only nearest neighbors also unlike the ldg method the cdg method works without stabilization for an arbitrary orientation of the element interfaces the computation of the numerical interface fluxes for the cdg method is slightly more involved than for the ldg method but this additional complication is clearly offset by increased compactness and flexibility compared to the br2 f bassi and s rebay j comput phys 131 1997 pp 267 279 and ip j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 lecture notes in phys 58 springer berlin 1976 pp 207 216 methods which are known to be compact the present method produces fewer nonzero elements in the matrix and is computationally more efficient

combined hybridizable discontinuous galerkin hdg and runge kutta discontinuous galerkin rk dg formulations for green naghdi equations on unstructured meshes abstract in this paper we introduce some new high order discrete formulations on general unstructured meshes especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations working with a recent family of optimized asymptotically equivalent equations we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlinear shallow water equations supplemented with several algebraic source terms which globally account for the non hydrostatic effects through the introduction of auxiliary coupling variables high order discrete approximations of the main flow variables are obtained with a rk dg method while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second order linear elliptic sub problems with high order hdg formulations the combined use of hybrid unknowns and local post processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches the proposed formulation is then extended to a more complex family of three parameters enhanced green naghdi equations the resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies showing good convergence properties and very good agreements with several sets of experimental data

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