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

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

optimal a priori error estimates for the hp version of the local discontinuous galerkin method for convection diffusion problems we study the convergence properties of the hp version of the local discontinuous galerkin finite element method for convection diffusion problems we consider a model problem in a one dimensional space domain we allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh width h in the polynomial degree p and in the regularity of the exact solution we identify a special numerical flux for which the estimates are optimal in both h and p the theoretical results are confirmed in a series of numerical examples

geometrical crossover and self similarity of dla and viscous fingering clusters abstract diffusion limited aggregation dla clusters and the structures observed in viscous fingering experiments at high capillary numbers are tree like fractals the different branches may be assigned a branch order in a way that exhibits scaling and permits a self similar characterization by the bifurcation ratio rn and the length ratio rl of branches of different orders the fractal dimension is given by d log r n log 1 r l good agreement between experiments and simulations is found a crossover function characterizes the branch orders and we conclude that dla is in a state of geometrical crossover branches are linear up to their length l but fractally distributed on length scales much larger than l the effective fractal dimension of dla depends on how d is measured and over what part of the cluster

superconvergence of the local discontinuous galerkin method for the linearized korteweg de vries equation we study the superconvergence property of the local discontinuous galerkin ldg method for solving the linearized korteweg de vries kdv equation we prove that if the piecewise p k polynomials with k 1 are used the ldg solution converges to a particular projection of the exact solution with the order k 3 2 when the upwind flux is used for the convection term and the alternating flux is used for the dispersive term numerical examples are provided at the end to support the theoretical results

superconvergence of the numerical traces of discontinuous galerkin and hybridized methods for convection diffusion problems in one space dimension in this paper we uncover and study a new superconvergence property of a large class of finite element methods for one dimensional convection diffusion problems this class includes discontinuous galerkin methods defined in terms of numerical traces discontinuous petrov galerkin methods and hybridized mixed methods we prove that the so called numerical traces of both variables superconverge at all the nodes of the mesh provided that the traces are conservative that is provided they are single valued in particular for is local discontinuous galerkin method we show that the superconvergence is order 2p 1 when polynomials of degree at most p are used extensive numerical results verifying our theoretical results are displayed

discontinuous galerkin error estimation for hyperbolic problems on unstructured triangular meshes we extend the error analysis of adjerid and baccouch 1 2 for the discontinuous galerkin discretization error to variable coefficient linear hyperbolic problems as well as nonlinear hyperbolic problems on unstructured meshes we further extend this analysis to transient hyperbolic problems and prove that the local superconvergence results presented in 1 still hold for both steady and transient variable coefficient linear and nonlinear problems this local error analysis allows us to construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on each element of general unstructured meshes we illustrate the superconvergence and the efficiency of our a posteriori error estimates by showing computational results for several linear and nonlinear numerical examples

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

nodal superconvergence of the local discontinuous galerkin method for singularly perturbed problems abstract in this paper a superconvergence of order ln n n 2 k 1 for the numerical traces of the ldg approximation to a one dimensional singularly perturbed convection diffusion reaction problem is proved the ldg method is applied on a shishkin mesh with 2 n elements and we use polynomials of degree at most k on each element this result puts the numerical finding reported in xie and zhang 2007 xie et al 2009 on firm mathematical ground

asymptotically exact a posteriori ldg error estimates for one dimensional transient convection diffusion problems in this paper new a posteriori error estimates for the local discontinuous galerkin ldg formulation applied to transient convection diffusion problems in one space dimension are presented and analyzed these error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element we first show that the leading error term on each element for the solution is proportional to a p 1 degree right radau polynomial while the leading error term for the solution s derivative is proportional to a p 1 degree left radau polynomial when polynomials of degree at most p are used these results are used to prove that for smooth solutions these error estimates at a fixed time converge to the true spatial errors in the l 2 norm under mesh refinement more precisely we prove that our ldg error estimates converge to the true spatial errors at o h p 5 4 rate finally we prove that the global effectivity indices in the l 2 norm converge to unity at o h 1 2 rate our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates

a discontinuous galerkin method for the wave equation abstract we present a new discontinuous galerkin method for solving the second order wave equation using the standard continuous finite element method in space and a discontinuous method in time directly applied to second order ode systems we prove several optimal a priori error estimates in space time norms for this new method and show that it can be more efficient than existing methods we also write the leading term of the local discretization error in terms of lobatto polynomials in space and jacobi polynomials in time which leads to superconvergence points on each space time cell we discuss how to apply our results to construct efficient and asymptotically exact a posteriori estimates for space time discretization errors numerical results are in agreement with theory

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

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

the discontinuous petrov galerkin method for elliptic problems abstract we propose a novel discontinuous mixed finite element formulation for the solution of second order elliptic problems fully discontinuous piecewise polynomial finite element spaces are used for the trial and test functions the discontinuous nature of the test functions at the element interfaces allows to introduce new boundary unknowns that on the one hand enforce the weak continuity of the trial functions and on the other avoid the need to define a priori algorithmic fluxes as in standard discontinuous galerkin methods static condensation is performed at the element level leading to a solution procedure based on the sole interface unknowns the resulting family of discontinuous dual primal mixed dpm finite element methods is presented in the one and two dimensional cases in the one dimensional case we show the equivalence of the method with implicit runge kutta schemes of the collocation type exhibiting optimal behavior and we develop the stability analysis numerical experiments in one and two dimensions demonstrate the order accuracy of the new method confirming the results of the analysis

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

discontinuous galerkin error estimation for linear symmetric hyperbolic systems in this manuscript we present an error analysis for the discontinuous galerkin discretization error of multi dimensional first order linear symmetric hyperbolic systems of partial differential equations we perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of legendre polynomials of degree p and p 1p 1 we apply these asymptotic results and solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error we present computational results for several linear hyperbolic systems in one two and three dimensions

approximate solution of the bagley torvik equation by hybridizable discontinuous galerkin methods in this paper we introduce a hybridizable discontinuous galerkin method for numerically solving a boundary value problem associated with the bagley torvik equation that arises in the study of the motion of a plate immersed in a newtonian fluid one of the main features of these methods is that they are efficiently implementable since it is possible to eliminate all internal degrees of freedom and obtain a global linear system that only involves unknowns at the element interfaces we display the results of a series of numerical experiments to ascertain the performance of the method

analysis of a family of discontinuous galerkin methods for elliptic problems the one dimensional case in this paper we analyze a family of discontinuous galerkin methods parameterized by two real parameters for elliptic problems in one dimension our main results are 1 a complete inf sup stability analysis characterizing the parameter values yielding a stable scheme and energy norm error estimates as a direct consequence thereof 2 an analysis of the error in l2 where the standard duality argument only works for special parameter values yielding a symmetric bilinear form and different orders of convergence are obtained for odd and even order polynomials in the nonsymmetric case the analysis is consistent with numerical results and similar behavior is observed in two dimensions

superconvergence of the local discontinuous galerkin method for the sine gordon equation in one space dimension abstract in this paper we present superconvergence results for the local discontinuous galerkin ldg method for the sine gordon nonlinear hyperbolic equation in one space dimension we identify a special numerical flux and a suitable projection of the initial conditions for the ldg scheme for which the l 2 norm of the ldg solution and its spatial derivative are of order p 1 when piecewise polynomials of degree at most p are used our numerical experiments demonstrate optimal order of convergence we further prove superconvergence toward particular projections of the exact solutions more precisely we prove that the ldg solution and its spatial derivative are o h p 3 2 super close to particular projections of the exact solutions while computational results show higher o h p 2 convergence rate our analysis is valid for arbitrary regular meshes and for p p polynomials with arbitrary p 1 numerical experiments validating these theoretical results are presented

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

singular perturbation analysis of time dependent convection diffusion equations in a circle abstract we study singularly perturbed time dependent convection diffusion equations in a circular domain considering suitable compatibility conditions we present convergence results and provide as well approximation schemes and error estimates to resolve the oscillations of classical numerical solutions due to the stiffness of our problem we construct via a specific boundary layer analysis the so called boundary layer elements which absorb the boundary layer singularities using a p1 classical finite element space enriched with the boundary layer elements we obtain an accurate numerical solution using a quasi uniform mesh that is without refinement of the mesh in the boundary layer

criticality in reactors under domain or external temperature perturbations the conditions for the onset of thermal runaway in reactors with small non uniformities is investigated the reaction is modelled by an arrhenius heat generation term with a finite activation energy and the dimensionless temperature u o is taken to satisfy a nonlinear equation of the form u 0 0f u 0 0 x d vu 0 bu 0 0 x d we investigate three classes of perturbations of this problem first we treat a small temperature variation maintained on the boundary of the domain secondly we consider a small distortion of the boundary of a circular cylindrical domain and thirdly we analyse the effect of a small hole in the domain in each case we derive asymptotic expansions for the critical frank kamenetskii parameter c where is a measure of the size of the perturbation a numerical scheme is then used to determine numerical values for the coefficients in the asymptotic expansion of c finally some of the asymptotic results are compared with corresponding numerical results obtained from a full numerical solution of the perturbed problem

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

uniform superconvergence analysis of the discontinuous galerkin method for a singularly perturbed problem in 1 d it has been observed from the authors numerical experiments 2007 that the local discontinuous galerkin ldg method converges uniformly under the shishkin mesh for singularly perturbed two point boundary problems of the convection diffusion type especially when using a piecewise polynomial space of degree k the ldg solution achieves the optimal convergence rate k 1 1 under the l 2 norm and a superconvergence rate 2k 1 1 for the one sided flux uniformly with respect to the singular perturbation parameter in this paper we investigate the theoretical aspect of this phenomenon under a simplified ode model in particular we establish uniform convergence rates 1nn n k 1 for the l 2 norm and ln n n 2k 1 for the one sided flux inside the boundary layer region here n even is the number of elements

a local discontinuous galerkin method for the second order wave equation abstract in this paper we present new superconvergence results for the local discontinuous galerkin ldg method applied to the second order scalar wave equation in one space dimension numerical experiments show o h p 1 l 2 convergence rate for the ldg solution and o hp 2 superconvergent solutions at radau points more precisely a local error analysis reveals that at a fixed time t the leading terms of the discretization errors for the solution and its derivative using p degree polynomial approximations are proportional to the p 1 degree right radau and p 1 degree left radau polynomials respectively thus the p degree ldg solution is o hp 2 superconvergent at the roots of the p 1 degree right radau polynomial and the derivative of the ldg solution is o hp 2 superconvergent at the roots of the p 1 degree left radau polynomial these results are used to construct simple efficient and asymptotically correct a posteriori error estimates in regions where solutions are smooth finally we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement

a superconvergence result for discontinuous galerkin methods applied to elliptic problems this paper presents a theoretical and numerical study of a class of discontinuous galerkin methods that shows the approximation of the gradient superconverges at the zeros of the legendre polynomials on a model 1d elliptic problem numerical experiments validate the theoretical results

numerical resolution of discontinuous galerkin methods for time dependent wave equations abstract the discontinuous galerkin dg method is known to provide good wave resolution properties especially for long time simulation in this paper using fourier analysis we provide a quantitative error analysis for the semi discrete dg method applied to time dependent linear convection equations with periodic boundary conditions we apply the same technique to show that the error is of order k 2 superconvergent at radau points on each element and of order 2 k 1 superconvergent at the downwind point of each element when using piecewise polynomials of degree k an analysis of the fully discretized approximation is also provided we compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes numerical results are provided to verify our error analysis

a uniformly convergent continuous discontinuous galerkin method for singularly perturbed problems of convection diffusion type abstract in this paper we introduce a coupled approach of local discontinuous galerkin and standard finite element method for solving singularly perturbed convection diffusion problems on shishkin mesh with linear elements a rate o n 1 ln n in an associated norm is established where n is the number of elements numerical experiments complement the theoretical results moreover a rate o n 2 ln 2 n in a discrete l norm and o n 2 in l2 norm are observed numerically on the shishkin mesh

a discontinuous galerkin method for higher order ordinary differential equations in this paper we propose a new discontinuous finite element method to solve initial value problems for ordinary differential equations and prove that the finite element solution exhibits an optimal o tp 1 convergence rate in the l2 norm we further show that the p degree discontinuous solution of differential equation of order m and its first m 1 derivatives are o t2p 2 m superconvergent at the end of each step we also establish that the p degree discontinuous solution is o tp 2 superconvergent at the roots of p 1 m degree jacobi polynomial on each step finally we present several computational examples to validate our theory and construct asymptotically correct a posteriori error estimates

accuracy enhancement of discontinuous galerkin solutions for convection diffusion equations in multiple dimensions discontinuous galerkin dg methods exhibit hidden accuracy that makes superconvergence of this method an increasing popular topic to address previous investigations have focused on the superconvergent properties of ordinary differential equations and linear hyperbolic equations additionally superconvergence of order k 3 2 for the convection diffusion equation that focuses on a special projection using the upwind flux was presented by cheng and shu in this paper we demonstrate that it is possible to extend the smoothness increasing accuracy conserving siac filter for use on the multidimensional linear convection diffusion equation in order to obtain 2k m order of accuracy where m depends upon the flux and takes on the values 0 1 2 or 1 the technique that we use to extract this hidden accuracy was initially introduced by cockburn luskin shu and suli for linear hyperbolic equations and extended by ryan et al as a smoothness increasing accuracy conserving filter we solve this convection diffusion equation using the local discontinuous galerkin ldg method and show theoretically that it is possible to obtain o h2k m in the negative order norm by post processing the ldg solution to a linear convection equation using a specially designed kernel such as the one by cockburn et al we can compute this same order accuracy in the l2norm additionally we present numerical studies that confirm that we can improve the ldg solution from o hk 1 to o h2k 1 using alternating fluxes and that we actually obtain o h2k 2 for diffusion dominated problems

locally conservative fluxes for the continuous galerkin method the standard continuous galerkin cg finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations a much needed quantity in many applications we show how to overcome this shortcoming by using a two step postprocessing the first step is the computation of a numerical flux trace defined on element interfaces and is motivated by the structure of the numerical traces of discontinuous galerkin methods this computation is nonlocal in that it requires the solution of a symmetric positive definite system but the system is well conditioned independently of mesh size so it can be solved at asymptotically optimal cost the second step is a local element by element postprocessing of the cg solution incorporating the result of the first step this leads to a conservative flux approximation with continuous normal components this postprocessing applies for the cg method in its standard form or for a hybridized version of it we present the hybridized version since it allows easy handling of variable degree polynomials and hanging nodes furthermore we provide an a priori analysis of the error in the postprocessed flux approximation and display numerical evidence suggesting that the approximation is competitive with the approximation provided by the raviart thomas mixed method of corresponding degree