a high order stable conservative method for solving hyperbolic conservation laws on arbitrarily distributed point clouds in this paper we aim to solve one and two dimensional hyperbolic conservation laws on arbitrarily distributed point clouds the initial condition is given on such a point cloud and the algorithm solves for point values of the solution at later time also on this point cloud by using the voronoi technique and by introducing a grouping algorithm we divide the computational domain into nonoverlapping cells each cell is a polygon and contains a minimum number of the given points to ensure accuracy we carefully select points in each cell during the grouping procedure and hence are able to interpolate or fit the discrete initial values with piecewise polynomials by adapting the traditional discontinuous galerkin method on the constructed polygonal mesh we obtain a stable conservative and high order method numerical results for both one and two dimensional scalar equations and euler systems of compressible gas dynamics are provided to illustrate the good behavior of our mesh generation algorithm as w

a time space flux corrected transport finite element formulation for solving multi dimensional advection diffusion reaction equations abstract we present a time space flux corrected transport fct finite element formulation for the multi dimensional time dependent advection diffusion reaction equation monotonic solutions can be achieved with the presented method while large time steps with courant number c r 1 are used numerical verification as well as a grid convergence analysis are carried out for 1d and 2d benchmark problems a 1d burgers equation with various reynolds numbers is solved with the time space fct method in addition a 2d transport problem with and without a nonlinear reaction inside of a cavity is considered finally the newly developed time space fct formulation is applied for modeling biofilm growth problems based on a continuum mathematical model it turns out that the time space fct method helps to reduce numerical dissipation and guarantees comparable numerical dispersion of the solution at the same time comparing to numerical solutions obtained by a time space finite incremental calculus fic method

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

first order least squares method with weakly imposed boundary condition for convection dominated diffusion problems we present and analyze a first order least squares method for convection dominated diffusion problems which provides robust l 2 a priori error estimate for the scalar variable even if the given data f l 2 the novel theoretical approach is to rewrite the method in the framework of discontinuous petrov galerkin dpg method and then show numerical stability by using a key equation discovered by gopalakrishnan and qiu 2014 this new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations we also show that the condition number of the global matrix is independent of the diffusion coefficient a key feature of the method is that there is no stabilization parameter chosen empirically in addition dirichlet boundary condition is weakly imposed numerical experiments verify our theoretical results and in particular show our way of weakly imposing dirichlet boundary condition is essential to the design of least squares methods numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes which are unstructured quasi uniform

stability analysis and a priori error estimates of the third order explicit runge kutta discontinuous galerkin method for scalar conservation laws in this paper we present an analysis of the runge kutta discontinuous galerkin method for solving scalar conservation laws where the time discretization is the third order explicit total variation diminishing runge kutta method we use an energy technique to prove the mathrm l 2 norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws quasi optimal order is obtained for general numerical fluxes and optimal order is given for upwind fluxes the theoretical results are obtained for piecewise polynomials with any degree k geq1 under the standard temporal spatial cfl condition tau leq gamma h where h and tau are the element length and time step respectively and the positive constant gamma is independent of h and tau

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

on discontinuous galerkin methods discontinuous galerkin methods have received considerable attention in recent years for problems in which advection and diffusion terms are present several alternatives for treating the diffusion and advective fluxes have been introduced this report summarizes some of the methods that have been proposed several numerical examples are included in the paper these present discontinuous galerkin solutions of one dimensional problems with a scalar variable results are presented for diffusion reaction problems and advection diffusion problems we discuss the performance of various formulations with respect to accuracy as well as stability of the method copyright 2003 john wiley sons ltd

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

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

quadrature free implementation of discontinuous galerkin method for hyperbolic equations a discontinuous galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions this approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent in the discontinuous galerkin method test problems include the linear and nonlinear one dimensional scalar advection of smooth initial value problems that are discretized by using unstructured grids with varying degrees of smoothness and regularity and two dimensional linear euler solutions on unstructured grids

the space continuous discontinuous galerkin method abstract the space discontinuous finite element method for a scalar diffusion convection reaction problem is introduced and analyzed in the solution method the boundary conditions and the continuity of the solution at the element interfaces are forced in a weak sense the convergence rate of the numerical solution is shown however to be the same as in the usual approach based on a continuous approximation finally the benefits of the method with discontinuous continuous approximation in h adaptation of the finite element space are illustrated with the aid of numerical calculations

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

adaptive motion compensated wavelet filtering for image sequence coding this paper deals with new advances made in the field of discrete spatio temporal filters applied to digital image sequences within time varying images the temporal correlation of the information is folded within the spatio temporal domain by motions originating from both camera and object displacements the spatio temporal video information can be therefore reformulated in terms of motion trajectories and intensity variations along these trajectories to yield that signal description the spatio temporal scenes will be segmented according to motion motion compensated temporal filters have been used as convolutional filters applied along the assumed motion trajectories spectral interpretations show the efficiency of motion compensated filtering for video signals as a matter of fact the whole signal analysis performed in this paper also includes a spatial filtering to achieve a complete spatio temporal 2 d t decomposition motion compensated filtering leads to multiresolution applications it leads to optimum and adaptive signal to noise decomposition procedures based on the temporal correlative content such properties allow enhancing tasks like temporal interpolation image sequence smoothing and restoration simulation results are presented in this paper to illustrate the field of image sequence coding

error estimate for the finite volume scheme applied to the advection equation we study the convergence of a finite volume scheme for the linear advection equation with a lipschitz divergence free speed in r d we prove a h 1 2 error estimate in the l infty 0 t l 1 norm for bv data this result was expected from numerical experiments and is optimal

a finite element method for the neutron transport equation in an infinite cylindrical domain we study the spatial discretization in a fully discrete scheme for the numerical solution of a model problem for the neutron transport equation in an infinite cylindrical domain based on using an interpolation technique in the discontinuous galerkin finite element procedure we derive an almost optimal error estimate for the scalar flux in the l2 norm combining a duality argument applied to the above result together with the previous semidiscrete error estimates for the velocity discretizations we also obtain globally optimal error bounds for the critical eigenvalues

a mass conservative characteristic fe scheme for optimal control problems governed by convection diffusion equations abstract in this paper we develop a mass conservative characteristic finite element scheme for optimal control problems governed by linear singularly perturbed convection diffusion equations we discuss the case that the velocity fields are non divergence free the space discretization of the state variable is done by piecewise linear continuous functions whereas the control variable is approximated by piecewise constant functions due to the limitation of the regularity the scheme preserves the mass balance for the original state equation we derive a priori error estimates for both the control and state approximations some numerical examples are presented to show the efficiency of the proposed scheme

a multilevel discontinuous galerkin method a variable v cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented an analysis under a mild regularity assumption shows that the preconditioner is uniform the interior penalty method is then combined with a discontinuous galerkin scheme to arrive at a discretization scheme for an advection diffusion problem for which an error estimate is proved a multigrid algorithm for this method is presented and numerical experiments indicating its robustness with respect to diffusion coefficient are reported

on a two level finite element method for the incompressible navier stokes equations we consider the galerkin finite element method for the incompressible navier stokes equations in two dimensions where the finite dimensional space s employed consist of piecewise polynomials enriched with residual free bubble rfb functions to find the bubble part of the solution a two level finite element method tlfem is described and its application to the navier stokes equation is displayed numerical solutions employing the tlfem are presented for three benchmark problems we compare the numerical solutions using the tlfem with the numerical solutions using a stabilized method

on the bruhat tits stratification of a quaternionic unitary rapoport zink space in this article we study the special fiber of the rapoport zink space attached to a quaternionic unitary group the special fiber is described using the so called bruhat tits stratification and is intimately related to the bruhat tits building of a split symplectic group as an application we describe the supersingular locus of the related shimura variety

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

positivity preserving high order local discontinuous galerkin method for parabolic equations with blow up solutions in this paper we apply positivity preserving local discontinuous galerkin ldg methods to solve parabolic equations with blow up solutions this model is commonly used in combustion problems however previous numerical methods are mainly based on a second order finite difference method this is because the positivity preserving property can hardly be satisfied for high order ones leading to incorrect blow up time and blow up sets recently we have applied discontinuous galerkin methods to linear hyperbolic equations involving singularities and obtained good approximations for nonlinear problems some special limiters are constructed to capture the singularities precisely we will continue this approach and study parabolic equations with blow up solutions we will construct special limiters to keep the positivity of the numerical approximations due to the dirichlet boundary conditions we have to modify the numerical fluxes and the limiters used in the schemes numerical experiments demonstrate that our schemes can capture the blow up sets and high order approximations yield better numerical blow up time

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

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

local error estimates for a finite element method for hyperbolic and convection diffusion equations local error estimates of near optimal order are derived for a finite element method for hyperbolic and convection dominated convection diffusion equations in a domain omega subset r 2 the method generates in an explicit fashion a continuous piecewise polynomial approximation of degree n geqslant 2 over a triangulation of omega the scheme is shown to propagate disturbances a distance o sqrt h log frac 1 h in the crosswind direction where h is the meshsize the analysis uses test functions which depend only on the crosswind variable it is also shown to be applicable in a parallel fashion to the discontinuous galerkin method thus underscoring the close interrelationship of the two methods

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

development of an adaptive discontinuous galerkin finite element method for advection reaction equations abstract this paper presents an adaptive discontinuous galerkin finite element method to solve linear advection reaction equations the method was developed for the prediction of hemolysis in prosthetic devices where the flow may include recirculation zones to ensure the well posedness of the problem we solve the transient form of the pde time discretization is achieved with an implicit euler scheme in order to obtain the steady state solution rapidly linear interpolation functions are used to approximate the hemolysis field the a posteriori estimation of the error is obtained by solving an advection reaction equation for the error which is approximated with quadratic interpolation functions then elementary estimations of the error are computed using a semi norm developed to achieve the best possible results even in the presence of recirculation code verification to assess the convergence and conservation properties of the method is performed with the method of manufactured solutions and with a conservation test using a sharp gaussian function as a reaction term excellent results are obtained in both cases predictions of hemolysis in a dialysis cannula are presented as an engineering application of our method

a finite element method for time dependent convection diffusion equations we present a finite element method for time dependent convection diffusion equations the method is explicit and is applicable with piecewise polynomials of degree n 2 in the limit of zero diffusion it reduces to a recently analyzed finite element method for hyperbolic equations near optimal error estimates are derived numerical results are given

a spacetime discontinuous galerkin method for scalar conservation laws abstract we present a spacetime discontinuous galerkin sdg finite element method for scalar hyperbolic conservation laws the method is consistent with the integral forms of the conservation law and its associated entropy condition written in terms of the physical godunov flux on each spacetime element the discrete basis functions are piecewise continuous in spacetime admitting discontinuities across element boundaries the resulting bubnov galerkin method is high order stable for both convex and non convex flux functions it does not require stabilization beyond the basic galerkin projection the sdg method is applicable to both layered and unstructured spacetime grids an element by element solution scheme delivers o n computational complexity where n is the number of elements when applied to spacetime meshes that conform to a special causality constraint we employ two different limiters to control the local overshoot and undershoot that the basic sdg projection generates near shocks

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

a vertex based scheme on polyhedral meshes for advection reaction equations with sub mesh stabilization we devise and analyze vertex based schemes on polyhedral meshes to approximate advection reaction equations error estimates of order o h3 2 are established in the discrete inf sup stability norm which includes the mesh dependent weighted advective derivative the two key ingredients are a local polyhedral reconstruction map leaving affine polynomials invariant and a local design of stabilization whereby gradient jumps are only penalized across some subfaces in the interior of each mesh cell numerical results are presented on three dimensional polyhedral meshes

an optimal order estimate for eulerian lagrangian localized adjoint methods for variable coefficient advection reaction problems previously we developed eulerian lagrangian localized adjoint methods ellams to solve advection reaction problems ellams provide a systematic approach to treat boundary conditions and to conserve mass and have proven to be powerful methods for advection dominated problems in this paper the authors conduct theoretical analysis for these ellams and prove that they have an optimal order convergence rate moreover the estimates involve only the derivatives of the exact solution in space and along the characteristics in contrast many existing methods have only suboptimal order error estimates for advection reaction problems and the estimates involve the temporal derivatives of the exact solution which are usually much larger than the derivatives along the characteristics key words advection reaction equations eulerian lagrangian method convergence analysis optimal order error estimates

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

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

hybrid methods for radiation transport using diagonally implicit runge kutta and space time discontinuous galerkin time integration abstract in this work we describe extensions of a hybrid method for time dependent linear kinetic radiation transport problems to high order time integration schemes of the diagonally implicit runge kutta dirk and space time discontinuous galerkin stdg types the hybrid methods are constructed by splitting the radiation flux into collided and uncollided components to which low and high resolution discrete ordinates approximations are applied the efficiency of hybrid methods constructed using dirk stdg integral deferred correction idc and implicit euler schemes is compared using a test problem in one dimensional slab geometry containing material discontinuities it is observed that i higher order methods are more efficient than the implicit euler method often by an order of magnitude or more ii third order methods yield solutions of a given error in roughly half the time of the second order method of the same type and iii for a given order of accuracy it is found that the most efficient class of time integration scheme is stdg followed by dirk with idc the least efficient for the test problem considered two test problems in two dimensional xy geometry are used to compare the computational efficiency of hybrid and standard discrete ordinates methods constructed with dirk and stdg integrators we observe that replacing a standard discrete ordinates method using an angular quadrature of order n with a hybrid discrete ordinates method using angular quadratures of order 2n and n 2 for the uncollided and collided fluxes respectively usually reduces overall solution time by a factor of 2 or more while simultaneously reducing the resulting solution error by a factor of 2 or more for the test problems considered

discontinuous galerkin method for krause s consensus models and pressureless euler equations in this paper we apply discontinuous galerkin dg methods to solve two model equations email protected s consensus models and pressureless euler equations these two models are used to describe the collisions of particles and the distributions can be identified as density functions if the particles are placed at a single point then the density function turns out to be a d function and is difficult to be well approximated numerically in this paper we use dg method to approximate such a singularity and demonstrate the good performance of the scheme since the density functions are always positive we apply a positivity preserving limiter to them moreover for pressureless euler equations the velocity satisfies the maximum principle we also construct special limiters to fulfill this requirement

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

numerical homogenization method for parabolic advection diffusion multiscale problems with large compressible flows we introduce a numerical homogenization method based on a discontinuous galerkin finite element heterogeneous multiscale method to efficiently approximate the effective solution of parabolic advection diffusion problems with rapidly varying coefficients large peclet number and compressible flows to estimate the missing data of an effective model numerical upscaling is performed which accurately captures the effects of microscopic solenoidal or gradient flow at a macroscopic scale such as enhancement or depletion of the effective diffusion for compressible flow with periodic data we derive sharp a priori error estimates for the macro and micro discretization errors which are robust in the advection dominated regime numerical tests confirm the error estimates for problems with periodic data and illustrate the applicability of our method for problems with non periodic data

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

an adaptive stabilized conforming finite element method via residual minimization on dual discontinuous galerkin norms abstract we design and analyze a new adaptive stabilized finite element method we construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf sup stability we formulate this residual minimization as a stable saddle point problem which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement numerical results on an advection reaction model problem show competitive error reduction rates when compared to discontinuous galerkin methods on uniformly refined meshes and smooth solutions moreover the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers

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

stabilized discontinuous galerkin methods for scalar linear hyperbolic equations abstract brezzi et al f brezzi l d marini e suli discontinuous galerkin methods for first order hyperbolic problems m 3 as 14 12 2004 1893 1903 have proposed stabilized discontinuous galerkin finite element methods to approximate the solution to scalar linear hyperbolic problems under a coercivity condition in this paper we relax such a condition stabilized dg methods are then applicable to more general problems arising in biology for example strong stability of the approximate solution is shown and optimal order a priori error estimates are obtained

on a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow the aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element fe problem for the approximation of viscoelastic fluid flow obeying an oldroyd b differential model the interest for this algorithm lies in its applications to numerical simulation and in the cost of computing furthermore it is easy to bring this algorithm into play the unknowns are sigma the viscoelastic part of the extra stress tensor u the velocity and p the pressure we suppose that the solution sigma u p is sufficiently smooth and small the approximation of stress velocity and pressure are resp p 1 discontinuous p 2 continuous p 1 continuous fe upwinding needed for convection of sigma is made by discontinuous fe the method consists to solve alternatively a transport equation for the stress and a stokes like problem for velocity and pressure previously results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm in this paper we show that the mapping of the decoupled fixed point algorithm is locally in a neighbourhood of sigma u p contracting and we obtain existence unicity locally of the solution of the approximate problem and error bounds

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

high order discontinuous galerkin methods for a class of transport equations with structured populations in this paper we analyze a discontinuous galerkin finite element method for approximating solutions to transport equations with certain nonlinearities we consider models for age structured populations allowing for a nonlinear removal rate with non local boundary conditions on the in flow boundary the method employs a stabilizing term over the interior edges allowing for convergence in a stronger than usual norm we establish convergence rates for general higher order basis functions and provide numerical examples consistent with this result

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

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

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability

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

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

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis