stability and error estimates of local discontinuous galerkin methods with implicit explicit time marching for advection diffusion problems the main purpose of this paper is to analyze the stability and error estimates of the local discontinuous galerkin ldg methods coupled with carefully chosen implicit explicit imex runge kutta time discretization up to third order accuracy for solving one dimensional linear advection diffusion equations in the time discretization the advection term is treated explicitly and the diffusion term implicitly there are three highlights of this work the first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the ldg methods the second is that by aid of the aforementioned relationship and the energy method we show that the imex ldg schemes are unconditionally stable for the linear problems in the sense that the time step tau is only required to be upper bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent

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

error analysis for optimal control problem governed by convection diffusion equations dg method in the current paper we study the convergence properties of the dgfe approximation of optimal control problem governed by convection diffusion equations we derive a posteriori error estimates and a priori error estimates for both the states ad joint and the control variable approximation for the optimal control problem these estimates are apparently not available in the literature

preconditioning methods for local discontinuous galerkin discretizations a multilevel interior penalty method is used as an efficient preconditioner for the schur complement of the local discontinuous galerkin ldg discretization of a poisson problem the method is then used in a block triangular preconditioner of the ldg saddle point system the block preconditioner is of the same efficiency as the schur complement version finally the block preconditioner is extended to the discretization of the stokes problem by the ldg method again the preconditioned saddle point problem can be solved in about as many steps as the schur complement the influence of several parameters on the performance of these methods is investigated

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

parabolic evolution equations in interpolation and extrapolation spaces on etudie le probleme de cauchy pour des equations d evolution lineaires a t u f t o t t de type parabolique dans un espace de banach x 0 on considere des resultats d existence d unicite et de continuite pour a t u u f t u o t t

optimal control of evolutionary quasi newtonian fluids we are interested in the necessary optimality conditions for optimal control problems of systems describing the unsteady flow of incompressible shear thickening fluids to overcome the difficulties related to the differentiability of the control to state mapping consequence of the lack of regularity of the state variable we first establish some intermediate optimality conditions and then pass to the limit

explicit runge kutta schemes and finite elements with symmetric stabilization for first order linear pde systems we analyze explicit runge kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first order linear differential operator in space of friedrichs type for the time discretization we consider explicit second and third order runge kutta schemes we identify a general set of properties on the space stabilization encompassing continuous and discontinuous finite elements under which we prove stability estimates using energy arguments then we establish l 2 norm error estimates with quasi optimal convergence rates for smooth solutions in space and time these results hold under the usual cfl condition for third order runge kutta schemes and any polynomial degree in space and for second order runge kutta schemes and first order polynomials in space for second order runge kutta schemes and higher polynomial degrees in space a tightened 4 3 cfl condition is required numerical results are presented for smooth and rough solutions the case of finite volumes is briefly discussed

error analysis of the semi discrete local discontinuous galerkin method for compressible miscible displacement problem in porous media in this paper local discontinuous galerkin method for flow and transport is introduced for the coupled system of compressible miscible displacement problem optimal error estimates in l 0 t l 2 norm not only for the solution itself but also for the auxiliary variables are derived the main technical difficulties in the analysis include the nonlinearity the coupling of the models and the treatment of the inter element jump terms which arise from the discontinuous nature of the numerical method the numerical results illustrate the accuracy and capability of the method

structural operators and eigenmanifold decomposition for functional differential equations in hilbert spaces this paper studies spectral properties of linear retarded functional differential equations in hilbert spaces with the emphasis on their relations to structural operators the equations involve unbounded operators acting on the discrete and distributed delayed terms and the operators acting on the instantaneous term are defined through sesquilinear forms the main concern of this paper is studying the spectral properties of the infinitesimal generators associated with the solution semigroups by means of structural operators the characterizations of eigenmanifolds are derived and the relations between the manifolds and structural operators are shown by using the properties of structural operators

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

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

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

stability analysis and error estimates of local discontinuous galerkin methods with implicit explicit time marching for nonlinear convection diffusion problems the main purpose of this paper is to analyze the stability and error estimates of the local discontinuous galerkin ldg methods coupled with implicit explicit imex time discretization schemes for solving one dimensional convection diffusion equations with a nonlinear convection both runge kutta and multi step imex methods are considered by the aid of the energy method we show that the imex ldg schemes are unconditionally stable for the nonlinear problems in the sense that the time step is only required to be upper bounded by a positive constant which depends on the flow velocity and the diffusion coefficient but is independent of the mesh size h we also give optimal error estimates for the imex ldg schemes under the same temporal condition if a monotone numerical flux is adopted for the convection numerical experiments are given to verify our main results

stabilized hp finite element methods for first order hyperbolic problems we analyze the hp version of the streamline diffusion finite element method sdfem and of the discontinuous galerkin finite element method dgfem for first order linear hyperbolic problems for both methods we derive new error estimates on general finite element meshes which are sharp in the mesh width h and in the spectral order p of the method assuming that the stabilization parameter is o h p for piecewise analytic solutions exponential convergence is established on quadrilateral meshes for the dgfem we admit very general irregular meshes and for the sdfem we allow meshes which contain hanging nodes numerical experiments confirm the theoretical results

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

on convergence of difference schemes for delay parabolic equations the convergence of difference schemes for the approximate solutions of the initial boundary value problem for the delay parabolic differential equation dv t dt av t b t v email protected t 0 v t g t email protected email protected 0 in an arbitrary banach space e with the unbounded linear operators a and b t in e with dense domains d a d b t is investigated theorems on convergence estimates for the solutions of the first and the second order of accuracy difference schemes in fractional spaces e a are established in practice the convergence estimates in holder norms for the solutions of difference schemes of the first and the second order of approximation in t of the approximate solutions of multi dimensional delay parabolic equations are obtained the theoretical statements for the solution of these difference schemes are supported by the numerical example

l2 regularity for parabolic partial integrodifferential equations with delay in the highest order derivatives abstract a partial integrodifferential equation is studied in which the derivatives of highest order also contain a discrete and a distributed delay by means of abstract regularity results global existence and uniqueness of a strict solution are obtained moreover a characterization of the infinitesimal generator of the solution operator is given

optimal control for an adsorbate induced phase transition model this paper is concerned with the optimal control problem for an adsorbate induced phase transition model that is we show the existence of the optimal control and derive the optimality conditions by showing the differentiability of the cost functional with respect to the control

optimality conditions for distributed control problems with nonlinear state equation we develop an abstract approximating process for control problems with convex cost governed by evolution equations involving monotone operators applications are given concerning necessary conditions of optimality for distributed control problems with hyperbolic parabolic and delay differential state systems these conditions are expressed by means of the clarke 11 generalized gradient

on a class of retarded partial differential equations on developpe des resultats recents sur des equations d evolution abstraites afin de les appliquer a un type d equations dans des espaces de banach avec un retard discret et continu ces equations peuvent etre considerees comme la version abstraite d equations integrodifferentielles partielles a retard

discontinuous galerkin methods for nonlinear scalar hyperbolic conservation laws divided difference estimates and accuracy enhancement in this paper an analysis of the accuracy enhancement for the discontinuous galerkin dg method applied to one dimensional scalar nonlinear hyperbolic conservation laws is carried out this requires analyzing the divided difference of the errors for the dg solution we therefore first prove that the alpha th order 1 le alpha le k 1 1 k 1 divided difference of the dg error in the l 2 l2 norm is of order k frac 3 2 frac alpha 2 k 32 2 when upwind fluxes are used under the condition that f u f u possesses a uniform positive lower bound by the duality argument we then derive superconvergence results of order 2k frac 3 2 frac alpha 2 2k 32 2 in the negative order norm demonstrating that it is possible to extend the smoothness increasing accuracy conserving filter to nonlinear conservation laws to obtain at least frac 3 2 k 1 32k 1 th order superconvergence for post processed solutions as a by product for variable coefficient hyperbolic equations we provide an explicit proof for optimal convergence results of order k 1 k 1 in the l 2 l2 norm for the divided differences of dg errors and thus 2k 1 2k 1 th order superconvergence in negative order norm holds numerical experiments are given that confirm the theoretical results

superconvergence of discontinuous galerkin and local discontinuous galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension in this paper we study the superconvergence property for the discontinuous galerkin dg and the local discontinuous galerkin ldg methods for solving one dimensional time dependent linear conservation laws and convection diffusion equations we prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection diffusion equations the order of superconvergence for both cases is proved to be k frac 3 2 when piecewise p k polynomials with k geq1 are used the proof is valid for arbitrary nonuniform regular meshes and for piecewise p k polynomials with arbitrary k geq1 improving upon the results in y cheng and c w shu j comput phys 227 2008 pp 9612 9627 y cheng and c w shu computers and structures 87 2009 pp 630 641 in which the proof based on fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise p 1 polynomials

first and second order optimality conditions for a class of optimal control problems with quasilinear elliptic equations a class of optimal control problems for quasilinear elliptic equations is considered where the coefficients of the elliptic differential operator depend on the state function first and second order optimality conditions are discussed for an associated control constrained optimal control problem main emphasis is laid on second order sufficient optimality conditions to this aim the regularity of the solutions to the state equation and its linearization is studied in detail and the pontryagin maximum principle is derived one of the main difficulties is the nonmonotone character of the state equation

discontinuous and coupled continuous discontinuous galerkin methods for the shallow water equations we consider the approximation of a simplified model of the depth averaged two dimensional shallow water equations by two approaches in both approaches a discontinuous galerkin dg method is used to approximate the continuity equation in the first approach a continuous galerkin method is used for the momentum equations in the second approach a particular dg method the nonsymmetric interior penalty galerkin method is used to approximate momentum a priori error estimates are derived and numerical results are presented for both approaches

error estimates for the third order explicit runge kutta discontinuous galerkin method for a linear hyperbolic equation in one dimension with discontinuous initial data in this paper we present an error estimate for the explicit runge kutta discontinuous galerkin method to solve a linear hyperbolic equation in one dimension with discontinuous but piecewise smooth initial data the discontinuous finite element space is made up of piecewise polynomials of arbitrary degree k ge 1 k 1 and time is advanced by the third order explicit total variation diminishing runge kutta method under the standard cfl temporal spatial condition the l 2 mathbb r backslash mathcal r t l 2 r r t norm error at the final time t t is optimal in both space and time where mathcal r t r t is the pollution region due to the initial discontinuity with the width mathcal o sqrt t beta h 1 2 log 1 h o t h 1 2 log 1 h here h h is the maximum cell length and beta is the flowing speed these results are independent of the time step and hold also for the semi discrete discontinuous galerkin method

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

superconvergence of the local discontinuous galerkin method for elliptic problems on cartesian grids in this paper we present a superconvergence result for the local discontinuous galerkin ldg method for a model elliptic problem on cartesian grids we identify a special numerical flux for which the l2 norm of the gradient and the l2 norm of the potential are of orders k 1 2 and k 1 respectively when tensor product polynomials of degree at most k are used for arbitrary meshes this special ldg method gives only the orders of convergence of k and k 1 2 respectively we present a series of numerical examples which establish the sharpness of our theoretical results

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

existence and continuous dependence results for nonlinear differential inclusions with infinite delay this paper is concerned with nonlinear functional differential inclusions with infinite delay in banach spaces using tools involving the measure of noncompactness and multi valued fixed point theory existence and continuous dependence results are obtained for integral solutions without the assumption of compactness on the associated nonlinear semigroup

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

optimal control of quasilinear parabolic equations this paper deals with optimal control problems governed by quasilinear parabolic equations in divergence form whose cost functional is of lagrangian type our aim is to prove the existence of solutions and derive some optimality conditions to attain this second objective we accomplish the sensitivity analysis of the state equation with respect to the control proving that under some assumptions this relation is g teaux differentiable finally a regularising procedure along with ekeland s variational principle allow us to treat some other problems for which this differentiability property cannot be stated

stability analysis of discontinuous galerkin discrete shock profiles for steady scalar conservation laws we present a stability analysis of stationary discrete shock profiles for a discontinuous galerkin method approximating scalar nonlinear hyperbolic conservation laws with a convex flux we consider the godunov method to evaluate the numerical flux and use either an explicit third order runge kutta scheme or an implicit backward euler scheme for the time integration applying a linear stability analysis we show that the steady solutions may become unstable when the numerical flux is not differentiable in particular the situation of a shock at an interface of the mesh corresponds to an unstable solution for a space discretization accuracy higher than second order whatever the time integration method and the physical flux spectra of the linearized operator indicate that the fourth order numerical scheme with the explicit time integration is also unstable for a continuous range of shock positions around an interface

error estimates for the runge kutta discontinuous galerkin method for the transport equation with discontinuous initial data we study the approximation of nonsmooth solutions of the transport equation in one space dimension by approximations given by a runge kutta discontinuous galerkin method of order two we take an initial datum which has compact support and is smooth except at a discontinuity and show that if the ratio of the time step size to the grid size is less than 1 3 the error at the time t in the l 2 mathbb r setminus mathcal r t norm is the optimal order two when mathcal r t is a region of size o t 1 2 h 1 2 log 1 h to the right of the discontinuity and of size o t 1 3 h 2 3 log 1 h to the left numerical experiments validating these results are presented

discontinuous galerkin method for hyperbolic equations involving delta singularities negative order norm error estimates and applications in this paper we develop and analyze discontinuous galerkin dg methods to solve hyperbolic equations involving delta singularities negative order norm error estimates for the accuracy of dg approximations to delta singularities are investigated we first consider linear hyperbolic conservation laws in one space dimension with singular initial data we prove that by using piecewise k th degree polynomials at time t the error in the h k 2 norm over the whole domain is k 1 2 th order and the error in the h k 1 mathbb r backslash mathcal r t norm is 2k 1 th order where mathcal r t is the pollution region due to the initial singularity with the width of order mathcal o h 1 2 log 1 h and h is the maximum cell length as an application of the negative order norm error estimates we convolve the numerical solution with a suitable kernel which is a linear combination of b splines to obtain l 2 error estimate of 2k 1 th order for the post processed solution moreover we also obtain high order superconvergence error estimates for linear hyperbolic conservation laws with singular source terms by applying duhamel s principle numerical examples including an acoustic equation and the nonlinear rendez vous algorithms are given to demonstrate the good performance of dg methods for solving hyperbolic equations involving delta singularities

optimal stabilization and time step constraints for the forward euler local discontinuous galerkin method applied to fractional diffusion equations abstract a time dependent model problem with the riesz or the riemann liouville fractional differential operator of order 1 2 is considered by penalyzing the primary variable of the minimal dissipation local discontinuous galerkin mdldg method with a term of order h 1 and using a von neumann analysis stability conditions proportional to h are derived for the forward euler method and both fractional operators in one dimensional domains the cfl condition is numerically studied with respect to the approximation degree and the stabilization parameter our analysis and computations carried out using explicit high order strong stability preserving runge kutta schemes reveal that the proposed penalization term is suitable for high order approximations and explicit time advancing schemes when is close to one a series of numerical experiments in 1d and 2d problems are presented to validate our theoretical results and those not covered by the theory

maximal regularity for evolution equations by interpolation and extrapolation on considere le probleme aux valeurs initiales pour l equation d evolution abstraite u t b t u t f t 0 t t ou b t est pour tout t le generateur infinitesimal d un semi groupe analytique dans un b espace e on considere le cas ou le domaine d b t de b t ne depend pas de t

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

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

discontinuous galerkin methods for solving elliptic variational inequalities we study discontinuous galerkin methods for solving elliptic variational inequalities of both the first and second kinds analysis of numerous discontinuous galerkin schemes for elliptic boundary value problems is extended to the variational inequalities we establish a priori error estimates for the discontinuous galerkin methods which reach optimal order for linear elements results from some numerical examples are reported

optimal control problems governed by some semilinear parabolic equations abstract this paper is concerned with the optimal control problem for the keller segel equations that is as a generalization of ryu and yagi j math anal appl 256 2001 45 66 we derive the optimality conditions for the optimal control problem governed by a semilinear abstract equation of nonmonotone type moreover we prove the uniqueness of the optimal control under some conditions

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

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

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

nonconvex optimal control of nonlinear monotone parabolic systems abstract an optimal control problem is studied for distributed systems governed by nonlinear parabolic pde s with state constraints the state equation is monotone in the state variable and nonlinear in the control variable the constraints and the cost functional are not necessarily convex relaxed controls are used to prove the existence of an optimal control moreover a minimum principle of relaxed optimality is established

stencil reduction algorithms for the local discontinuous galerkin method the problem of reducing the stencil of the local discontinuous galerkin method applied to second order differential operator is discussed heuristic algorithms to minimize the total number of non zero blocks of the reduced stiffness matrix are presented and tested on a wide variety of unstructured and structured grids in 2d and 3d copyright 2009 john wiley sons ltd

hp fem for reaction diffusion equations i robust exponential convergence a singularly perturbed reaction diffusion equation in two dimensions is considered we assume analyticity of the input data i e the boundary of the domain is an analytic curve and the right hand side is analytic we show that the hp version of the finite element method leads to robust exponential convergence provided that one layer of needle elements of width o p eps is inserted near the domain boundary that is the rate of convergence is o mbox rm exp b p and independent of the perturbation parameter eps additionally we show that the use of numerical quadrature for the evaluation of the stiffness matrix and the load vector retains the exponential rate of convergence in particular the spectral element method based on the use of a gauss lobatto quadrature rule with p 1 x p 1 points yields robust exponential convergence

solvability of the cauchy problem for infinite delay equations abstract in this paper we study the solvability of the cauchy problem for functional differential equations with infinite delay in a banach space x which are more general than those in many previous publications we undertake our study under suitable hypotheses based on noncompactness measures and kamke type functions and in a general setting of admissible phase space p including the space lp 0 x therefore our results extend and unify many existing results for finite and infinite delay equations as a sample of possible applications we analyze the solvability of the cauchy problem for a schrodinger type partial functional integro differential equation in l1 r

abstract parabolic problems with parameter and application in this work the uniform well posedenes of singular perturbation problems for parameter dependent parabolic differential operator equations is established these problems occur in phytoremediation modelling

a local discontinuous galerkin method for the burgers poisson equation in this work we design analyze and test a local discontinuous galerkin method for solving the burgers poisson equation this model proposed by whitham linear and nonlinear waves 1974 as a simplified model for shallow water waves admits conservation of both momentum and energy as two invariants the proposed numerical method is high order accurate and preserves two invariants hence producing solutions with satisfying long time behavior the l 2 l 2 stability of the scheme for general solutions is a consequence of the energy preserving property the optimal order of accuracy for polynomial elements of even degree is proven a series of numerical tests is provided to illustrate both accuracy and capability of the method

on the optimal control of strongly nonlinear evolution equations abstract in this paper we study optimal control problems governed by nonlinear evolution equations with nonmonotone nonlinearities in the state first we prove an existence theorem then using the penalty method we obtain necessary conditions for optimality from them we derive a bang bang characterization of optimal controls finally an example of a nonlinear parabolic optimal control system is worked out in detail

coupling of continuous and discontinuous galerkin methods for transport problems in this paper we formulate a coupled discontinuous continuous galerkin method for the numerical solution of convection diffusion transport equations where convection may be dominant one motivation for this approach is to use a discontinuous method where the solution is rough e g in regions of high gradients and use a continuous method where the solution is smooth in this approach the domain is decomposed into two regions and appropriate transmission conditions are applied at the interface between regions in one region a local discontinuous galerkin method is applied and in the other region a standard continuous galerkin method is employed stability and a priori error estimates for the coupled method are derived and numerical results in one space dimension are given for smooth problems and problems with sharp fronts 2002 elsevier science b v all rights reserved

distributed control of systems governed by a general class of quasilinear elliptic equations abstract in this paper we are concerned with optimal control problems governed by a general class of quasilinear elliptic equations our aim is to prove existence of solution and derive optimality conditions in order to deal with this last question we begin investigating differentiability of state solution of equation with respect to the control second term of equation obtaining it only in certain cases a perturbation method is used to treat other cases as a consequence of the optimality system regularity of optimal control and state is establised

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

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

a discontinuous galerkin front tracking scheme and its optimal optimal error estimation an error estimate of optimal convergence rates and optimal error propagation optimal optimal was given for the numerical solutions produced by the runge kutta discontinuous galerkin rkdg method on the scalar nonlinear conservation laws in the case of smooth solutions in sun and rumsey 2013 this manuscript generalizes the problem to the case of a piecewise smooth solution containing one fully developed shock a front tracking technique is incorporated in the rkdg scheme to produce a numerical solution with a truly high order error the numerical smoothness approach of sun and rumsey 2013 is generalized to this particular case of a discontinuous solution