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

a hybridizable discontinuous galerkin method for stokes flow abstract in this paper we introduce a hybridizable discontinuous galerkin method for stokes flow the method is devised by using the discontinuous galerkin methodology to discretize a velocity pressure gradient formulation of the stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization one of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries thereby leading to a significant reduction in the size of the resulting matrix moreover by using an augmented lagrangian method the globally coupled unknowns are further reduced to the numerical trace of the velocity only another important feature is that the approximations of the velocity pressure and gradient converge with the optimal order of k 1 in the l 2 norm when polynomials of degree k 0 are used to represent the approximate variables based on the optimal convergence of the hdg method we apply an element by element postprocessing scheme to obtain a new approximate velocity which converges with order k 2 in the l 2 norm for k 1 the postprocessing performed at the element level is less expensive than the solution procedure numerical results are provided to assess the performance of the method

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

numerical study of the skin friction on a spheroid at incidence calcul numerique du champ d ecoulement tridimensionnel incompressible laminaire les equations sont simplifiees en negligeant les termes de diffusion pour obtenir les equations de navier stokes parabolisees

an anisotropic a priori error analysis for a convection dominated diffusion problem using the hdg method abstract this paper deals with the a priori error analysis for a convection dominated diffusion 2d problem when applying the hdg method on a family of anisotropic triangulations it is known that in this case boundary or interior layers may appear therefore it is important to resolve these layers in order to recover if possible the expected order of approximation in this work we extend the use of hdg method on anisotropic meshes in this context when the discrete local spaces are polynomials of degree k 0 this approach is able to recover an order of convergence k 1 2 in l 2 for all the variables under certain assumptions on the stabilization parameter and family of triangulations numerical examples confirm our theoretical results

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

analysis and development of adjoint based h adaptive direct discontinuous galerkin method for the compressible navier stokes equations abstract in this paper an adjoint based high order h adaptive direct discontinuous galerkin method is developed and analyzed for the two dimensional steady state compressible navier stokes equations particular emphasis is devoted to the analysis of the adjoint consistency for three different direct discontinuous galerkin discretizations including the original direct discontinuous galerkin method ddg the direct discontinuous galerkin method with interface correction ddg ic and the symmetric direct discontinuous galerkin method sddg theoretical analysis shows the extra interface correction term adopted in the ddg ic method and the sddg method plays a key role in preserving the adjoint consistency to be specific for the model problem considered in this work we prove that the original ddg method is not adjoint consistent while the ddg ic method and the sddg method can be adjoint consistent with appropriate treatment of boundary conditions and correct modifications towards the underlying output functionals the performance of those three ddg methods is carefully investigated and evaluated through typical test cases based on the theoretical analysis an adjoint based h adaptive ddg ic method is further developed and evaluated numerical experiment shows its potential in the applications of adjoint based adaptation for simulating compressible flows

multigrid acceleration of a relaxation procedure for the reduced navier stokes equations on applique la methode multigrille pour ameliorer la convergence d une methode de relaxation iterative pour la resolution numerique des equations de navier stokes reduites

numerical solution of incompressible flows by a marching multigrid nonlinear method on decrit un schema iteratif de resolution des equations de navier stokes pour un systeme de coordonnees orthogonales curvilignes l algorithme propose est independant du nombre de reynolds et peut etre applique a la resolution des equations d euler incompressibles

solution to the linear bgk equation via the principles of invariance the solution to the linearized bgk equation of kinetic theory is obtained via the principles of invariance as formulated by chandrasekhar the solution represents an alternative to the singular eigenfunction method featuring regular integral equations which are more easily evaluated laplace transforms are obtained for the net flow velocity and velocity distribution for both the albedo and milne problems with wall reflection the transforms are then numerically inverted to a high degree of accuracy results are compared to those of kainz and titulaer

a method for solving three dimensional viscous incompressible flows over slender bodies abstract a marching iterative method for solving the three dimensional incompressible and steady reduced navier stokes equations in general orthogonal coordinate systems is described with the velocity and the pressure as dependent variables the coupled set of the linearized finite difference continuity and momentum equations are solved iteratively without any splitting or factorization errors each iteration consists of spatial marching from the upstream boundary to the downstream boundary the discrete continuity and the two linearized crossflow momentum equations are satisfied at each marching step even when the mainstream momentum equation is not converged this solution procedure is equivalent to the solution of a single poisson like equation by the successive plane over relaxation method while other available solution methods employ a jacobi type iterative scheme and therefore are less efficient several properties of the numerical method have been assessed through a series of tests performed on the laminar incompressible flow over prolate spheroids at intermediate incidence

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

dual scale galerkin methods for darcy flow abstract the discontinuous galerkin dg method has found widespread application in elliptic problems with rough coefficients of which the darcy flow equations are a prototypical example one of the long standing issues of dg approximations is the overall computational cost and many different strategies have been proposed such as the variational multiscale dg method the hybridizable dg method the multiscale dg method the embedded dg method and the enriched galerkin method in this work we propose a mixed dual scale galerkin method in which the degrees of freedom of a less computationally expensive coarse scale approximation are linked to the degrees of freedom of a base dg approximation we show that the proposed approach has always similar or improved accuracy with respect to the base dg method with a considerable reduction in computational cost for the specific definition of the coarse scale space we consider raviart thomas finite elements for the mass flux and piecewise linear continuous finite elements for the pressure we provide a complete analysis of stability and convergence of the proposed method in addition to a study on its conservation and consistency properties we also present a battery of numerical tests to verify the results of the analysis and evaluate a number of possible variations such as using piecewise linear continuous finite elements for the coarse scale mass fluxes

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

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

a gaussian quadrature procedure for use in the solution of the boltzmann equation and related problems abstract a new gaussian quadrature procedure is developed for integrals of the form 0 e y 2 y p y dy for p 0 1 and 2 recursion relations are derived for the coefficients in the general three term recurrence relation for the polynomials whose roots are the quadrature abscissae a comparison with the gauss laguerre quadrature procedure is presented solutions of the chemical kinetic boltzmann equation are obtained with a discrete ordinate method based on this gaussian quadrature procedure the results are compared with previous solutions obtained with a polynomial expansion method

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

conditions for superconvergence of hdg methods for second order elliptic problems we provide a projection based analysis of a large class of finite element methods for second order elliptic problems it includes the hybridized version of the main mixed and hybridizable discontinuous galerkin methods the main feature of this unifying approach is that it reduces the main difficulty of the analysis to the verification of some properties of an auxiliary locally defined projection and of the local spaces defining the methods sufficient conditions for the optimal convergence of the approximate flux and the superconvergence of an element by element postprocessing of the scalar variable are obtained new mixed and hybridizable discontinuous galerkin methods with these properties are devised which are defined on squares cubes and prisms

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

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

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 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

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

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

conservative high order particle mesh scheme with applications to advection dominated flows abstract by combining concepts from particle in cell pic and hybridized discontinuous galerkin hdg methods we present a particle mesh scheme for flow and transport problems which allows for diffusion free advection while satisfying mass and momentum conservation locally and globally and extending to high order spatial accuracy this is achieved via the introduction of a novel particle mesh projection operator which casts the particle mesh data transfer as a pde constrained optimization problem permitting advective flux functionals at cell boundaries to be inferred from particle trajectories this optimization problem seamlessly fits in a hdg framework whereby the control variables in the optimization problem serve as advective fluxes in the hdg scheme the resulting algebraic problem can be solved efficiently using static condensation the performance of the scheme is demonstrated by means of numerical examples for the linear advection diffusion equation and the incompressible navier stokes equations the results demonstrate optimal spatial accuracy and when combined with a time integration scheme second order temporal accuracy is shown

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 parallel and adaptive hybridized discontinuous galerkin method for anisotropic nonhomogeneous diffusion abstract the diffusion equation in anisotropic and nonhomogeneous media arises in the study of flow through porous media with sharp material interfaces we discuss the solution of this problem by a hybrid discontinuous galerkin hdg method the method can be applied in three steps first we use a condensation technique to derive the scalar variable and the flux inside each element in terms of the numerical trace on the faces of that element then we form a global system of equations to solve for these numerical traces we then solve the equation inside each element for the internal unknowns using the obtained numerical traces in the global solve step similar to other dg variants hdg is a locally conservative method and a noticeable share of calculations are performed independently within each element in a mesh with p th order elements p 0 this method gives p 1 order of accuracy for smooth solutions for both the scalar variable and the flux moreover by using a simple post processing technique one can reach an accuracy of order p 2 for the scalar variable to be able to handle problems with sharp material discontinuities we use an adaptive refinement strategy with octree grid structure hence we avoid a larger global system which would arise from a fine uniform grid in this process we refine those elements with highest gradient of flux at two sides of their faces we have also utilized shared and distributed memory parallelism to enhance the performance of the method the method is implemented using different modules of deal ii bangerth and kanschat 1999 petsc balay et al 2015 p4est burstedde et al 2011 and hypre falgout and yang 2002 to demonstrate the accuracy efficiency scalability and flexibility of the method several two and three dimensional numerical experiments are studied

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

a superconvergent hybridizable discontinuous galerkin method for dirichlet boundary control of elliptic pdes we begin an investigation of hybridizable discontinuous galerkin hdg methods for approximating the solution of dirichlet boundary control problems governed by elliptic pdes these problems can involve atypical variational formulations and often have solutions with low regularity on polyhedral domains these issues can provide challenges for numerical methods and the associated numerical analysis we propose an hdg method for a dirichlet boundary control problem for the poisson equation and obtain optimal a priori error estimates for the control specifically under certain assumptions for a 2d convex polygonal domain we show the control converges at a superlinear rate we present 2d and 3d numerical experiments to demonstrate our theoretical results

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

the structure of the stationary kinetic boundary layer for the linear bgk equation recently very accurate numerical methods for solving kinetic boundary layer problems for linear kinetic equations have been developed to test such methods detailed information about exactly solvable models can be very helpful to this purpose the authors present results for the stationary 1d linear bgk equation obtained by the singular eigenfunction method that extend results already known in the literature both in scope and in precision special attention is paid to the nature of the singularity in the distribution function for small velocities near the boundary this singularity cannot be reproduced exactly by the numerical methods available explicit expressions are presented for the milne problem and for the albedo problem with input particles of a single velocity they compare the results with those of a recently developed variant of the two stream moment method they find excellent agreement except close to the singularity for many quantities of interest the accuracy obtainable by the novel two stream moment method cannot be reproduced with comparable numerical effort by evaluation of the exact solution

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

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

high order exactly divergence free hybrid discontinuous galerkin methods for unsteady incompressible flows abstract in this paper we present an efficient discretization method for the solution of the unsteady incompressible navier stokes equations based on a high order hybrid discontinuous galerkin formulation the crucial component for the efficiency of the discretization method is the distinction between stiff linear parts and less stiff non linear parts with respect to their temporal and spatial treatment exploiting the flexibility of operator splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub problems a corresponding hyperbolic transport problem and an unsteady stokes problem for the hyperbolic transport problem a spatial discretization with an upwind discontinuous galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation the treatment of the stokes part involves the solution of linear systems in this case a discretization with hybrid discontinuous galerkin methods is better suited we consider such a discretization for the stokes part with two important features h div conforming finite elements to guarantee exactly divergence free velocity solutions and a projection operator which reduces the number of globally coupled unknowns we present the method discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems

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

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

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

the discontinuous enrichment method for elastic wave propagation in the medium frequency regime the discontinuous enrichment method dem is specified and developed for the solution of two dimensional elastic wave propagation problems in the frequency domain the classical finite element polynomial approximation of the displacement field is enriched by the superposition of discontinuous pressure and shear wave functions the continuity of the solution across the element interfaces is weakly enforced by suitable lagrange multipliers higher order rectangular dem elements are constructed and benchmarked against the standard higher order polynomial galerkin elements for two dimensional problems in the medium frequency regime in general it is found that for such applications dem can achieve the same accuracy as the p finite element method using a similar computational complexity but with about 10 times fewer degrees of freedom this highlights the potential of this hybrid method for problems where the scale of vibrations is very small compared to the characteristic dimension of the physical medium copyright 2006 john wiley sons ltd

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 discontinuous enrichment method for variable coefficient advection diffusion at high peclet number a discontinuous galerkin method with lagrange multipliers is presented for the solution of variable coefficient advection diffusion problems at high peclet number in this method the standard finite element polynomial approximation is enriched within each element with free space solutions of a local constant coefficient homogeneous counterpart of the governing partial differential equation hence in the two dimensional case the enrichment functions are exponentials each exhibiting a sharp gradient in a carefully chosen flow direction the continuity of the enriched approximation across the element interfaces is enforced weakly by the aforementioned lagrange multipliers numerical results obtained for two benchmark problems demonstrate that elements based on the proposed discretization method are far more competitive for variable coefficient advection diffusion analysis in the high peclet number regime than their standard galerkin and stabilized finite element comparables copyright 2010 john wiley sons ltd

an accurate two stream moment method for kinetic boundary layer problems of linear kinetic equations the authors develop a method for solving 1d kinetic boundary layer problems for linear equations that uses separate hermite moment expansions in the velocity variable for particles moving towards and away from a plane wall this so called two stream method is tested for two especially simple kinetic equations the linear bgk equation and the klein kramers equation the kinetic equation for brownian particles for both of these equations extensive exact information is available for two simple boundary layer problems the milne and the albedo problem a comparison of these exact results with those obtained with this version of the two stream moment method shows that the accuracy obtainable by this method exceeds that of earlier methods by several orders of magnitude in particular the method allows them to obtain accurate results not only for the moments of the distribution function but also for the distribution function itself

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

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

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 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

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

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

discontinuous galerkin approximation of two phase flows in heterogeneous porous media with discontinuous capillary pressures we design and investigate a sequential discontinuous galerkin method to approximate two phase immiscible incompressible flows in heterogeneous porous media with discontinuous capillary pressures the nonlinear interface conditions are enforced weakly through an adequate design of the penalties on interelement jumps of the pressure and the saturation an accurate reconstruction of the total velocity is considered in the raviart thomas nedelec finite element spaces together with diffusivity dependent weighted averages to cope with degeneracies in the saturation equation and with media heterogeneities the proposed method is assessed on one dimensional test cases exhibiting rough solutions degeneracies and capillary barriers stable and accurate solutions are obtained without limiters

optimization of a regularized distortion measure to generate curved high order unstructured tetrahedral meshes we present a robust method for generating high order nodal tetrahedral curved meshes the approach consists of modifying an initial linear mesh by first introducing high order nodes second displacing the boundary nodes to ensure that they are on the cad surface and third smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high order mesh the smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh this means that whenever possible the resulting mesh preserves the geometrical features of the initial linear mesh such as shape stretching and size we present several examples to illustrate the performance of the proposed algorithm furthermore the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements we consider cases with polynomial approximations up to degree ten large deformations of the curved boundaries concave boundaries and highly stretched boundary layer elements the meshes obtained are suitable for high order finite element analyses

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

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

adaptive mixed hybrid and penalty discontinuous galerkin method for two phase flow in heterogeneous media in this paper we present a hybrid method which consists of a mixed hybrid finite element method and a penalty discontinuous galerkin method for the approximation of a fractional flow formulation of a two phase flow problem in heterogeneous media with discontinuous capillary pressure the fractional flow formulation is comprised of a wetting phase pressure equation and a wetting phase saturation equation which are coupled through a total velocity and the saturation affected coefficients for the wetting phase pressure equation the continuous mixed hybrid finite element method space can be utilized due to a fundamental property that the wetting phase pressure is continuous while it can reduce the computational cost by using less degrees of freedom and avoiding the post processing of velocity reconstruction this method can also keep several good properties of the discontinuous galerkin method which are important to the fractional flow formulation such as the local mass balance continuous normal flux and capability of handling the discontinuous capillary pressure for the wetting phase saturation equation the penalty discontinuous galerkin method is utilized due to its capability of handling the discontinuous jump of the wetting phase saturation furthermore an adaptive algorithm for the hybrid method together with the centroidal voronoi delaunay triangulation technique is proposed five numerical examples are presented to illustrate the features of proposed numerical method such as the optimal convergence order the accurate and efficient velocity approximation and the applicability to the simulation of water flooding in oil field and the oil trapping or barrier effect phenomena

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