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

on the poincare friedrichs inequality for piecewise h 1 functions in anisotropic discontinuous galerkin finite element methods the purpose of this paper is to propose a proof for the poincare friedrichs inequality for piecewise h 1 functions on anisotropic meshes by verifying suitable assumptions involved in the newly proposed proof we show that the poincare friedrichs inequality for piecewise h 1 functions holds independently of the aspect ratio which characterizes the shape regular condition in finite element analysis in addition under the maximum angle condition we establish the poincare friedrichs inequality for the crouzeix raviart non conforming linear finite element counterexamples show that the maximum angle condition is only sufficient

kernel sections and uniform attractors of multi valued semiprocesses abstract we present the existence of kernel sections which are all compact invariant and pullback attracting of an infinite dimensional general multi valued process constructed by the set valued backward extension of multi valued semiprocesses moreover the structure of the uniform attractors of a family of multi valued semiprocesses and the uniform forward attraction of kernel sections of a family of general multi valued processes are investigated finally we explain our abstract results by considering the mixed wave systems with supercritical exponent and ordinary differential equations

a robust wg finite element method for convection diffusion reaction equations this paper proposes and analyzes a weak galerkin wg finite element method for 2 and 3 dimensional convection diffusion reaction problems on conforming or nonconforming polygon polyhedral meshes the wg method uses piecewise polynomial approximations of degrees k k 0 for both the scalar function and its trace on the inter element boundaries we show that the method is robust in the sense that the derived a priori error estimates is uniform with respect to the coefficients for sufficient smooth true solutions numerical experiments confirm the theoretical results

discontinuous galerkin method based on peridynamic theory for linear elasticity modeling of discontinuities shock waves crack surfaces etc in solid mechanics is one of the major research areas in modeling the mechanical behavior of materials among the numerical methods the discontinuous galerkin method dgm poses some advantages in solving these problems in this study a novel formulation for dgm is derived for elastostatics based on the peridynamic theory derivation of the proposed formulation is presented numerical analyses are performed for different problems and the numerical results are compared to that of the known exact solutions of the problems the proposed weak formulation is stable and coercive peridynamic discontinuous galerkin formulation is found to be robust and successful in modeling elastostatic problems copyright 2011 john wiley sons ltd

discontinuous galerkin finite element approximation of the cahn hilliard equation with convection the paper is concerned with the construction and convergence analysis of a discontinuous galerkin finite element method for the cahn hilliard equation with convection using discontinuous piecewise polynomials of degree p geq1 and backward euler discretization in time we show that the order parameter c is approximated in the broken rm l infty rm h 1 norm with optimal order cal o h p tau the associated chemical potential w phi c gamma 2 delta c is shown to be approximated with optimal order cal o h p tau in the broken rm l 2 rm h 1 norm here phi c frac 1 4 1 c 2 2 is a quartic free energy function and gamma 0 is an interface parameter numerical results are presented with polynomials of degree p 1 2 3

numerical multiscale methods for a reaction dominated model a galerkin enriched finite element method gem is proposed for the singularly perturbed reaction diffusion equation this new method is an improvement on the petrov galerkin enriched method pgem where now the standard piecewise bi linear test space incorporates fine scales this appears as the fundamental ingredient for suppressing oscillations in the numerical solutions also new parameter free stabilized finite element methods derived from both the gem and the pgem are driven by local generalized eigenvalue problems in the process jump stabilizing terms belonging to the class of cip methods emerge as a result of the enriching procedure interestingly numerical results indicate that jump based stabilizations are unnecessary and sometimes undesirable when treating reaction dominated problems finally we establish relationships with more standard enriched and stabilized methods and show that the proposed methods outperform them numerically

upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation abstract for a smooth bounded domain r n n 3 and e a real parameter consider the hyperbolic equation eu tt u t u u g in with dirichlet boundary conditions under certain conditions on this equation has a compact attractor a e in h 0 1 l 2 for e 0 the parabolic equation also has a compact attractor which can be naturally embedded into a compact set a 0 in h 0 1 l 2 it is shown that for any neighborhood u of a 0 the set a e u for e small

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity motivation formulation and numerical examples the goal of this paper is to motivate introduce and demonstrate a novel approach to stabilizing discontinuous galerkin dg methods in nonlinear elasticity problems the stabilization term adapts to the solution of the problem by locally changing the size of a penalty term on the appearance of discontinuities with the goal of better approximating the solution consequently it is called an adaptive stabilization strategy the need for such a strategy is motivated through two and three dimensional examples in nonlinear elasticity the proposed scheme is simple to implement and compute and its performance is demonstrated with two and three dimensional numerical examples the accuracy of the proposed method is compared against a conforming method of the same order and a dg method with a traditional form of stabilization results for trilinear hexahedral elements indicate that the new stabilization strategy is more robust and more accurate when compared to a traditional form of stabilization however conforming trilinear hexahedral elements proved to be more computationally efficient for the examples shown here a two dimensional example with linear triangular elements showed comparable performances between the proposed method and a conforming one

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

upper semicontinuity of attractors for approximations of semigroups and partial differential equations abstract suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite dimensional system conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined applications are given to parabolic and hyperbolic partial differential equations

bubble stabilization of discontinuous galerkin methods we analyze the stabilizing effect of the introduction of suitable bubble functions in dg formulations for linear second order elliptic problems working for the sake of simplicity on laplace operator in particular we find that the addition of a single b ubble per element can stabilize the non symmetric formulation of baumann oden most commonly used dg methods need the addition of suitable stabilizing terms in order to provide good convergence properties the typical stabilizing procedure consists in the introduction of penalty terms that penalize the jumps across neigh boring elements sometimes in hyperbolic or in convection dominated problems one also use upwind techniques consisting in replacing the average u u 2 on an internal edge with the upwind value that is u or u according with the direction of the wind this however in most cases can be seen again as a jump stabilization 15 13 10 another possible way of stabilizing dg methods consists in the addition of suit able terms this time internal to each element of the so ca lled hughes francatype in general the integral of the original equation or one of t he original equations written in strong form inside each element in terms of the finite element unknowns trial functions multiplied by a similar expression act ing on the test functions the most famous stabilization of this type for standard galerkin methods is surely the supg stabilization of convection dominated equations 12 a typical problem in these cases is the choice of the proper stabilization coefficient to be put in front of the stabilizing term in a recent paper see 6 we pointed out that in dg methods the jumps are themselves to be regarded as equations so that jump stabilizations and hence upwind could be regarded as hughes franca stabilizations as well and indeed the optimal choice of the coefficient in a jump stabilizatio n term is still a subject that might need a further investigation

quasi optimal convergence rate of an adaptive discontinuous galerkin method we analyze an adaptive discontinuous finite element method adfem for symmetric second order linear elliptic operators the method is formulated on nonconforming meshes made of simplices or quadrilaterals with any polynomial degree and in any dimension geq2 we prove that the adfem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops we design a refinement procedure that maintains the level of nonconformity uniformly bounded and prove that the approximation classes using continuous and discontinuous finite elements are equivalent the geometric decay and the equivalence of classes are instrumental in deriving the optimal cardinality of the adfem we show that the adfem and the afem on nonconforming meshes yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

reduced integration post processing techniques for darcy s velocity recovery in bilinear quadrilaterals in this work we propose the use of a reduced integration technique for bilinear quadrilateral elements in the determination of the darcy s velocity via global and local post processing techniques this approach increases accuracy in distorted meshes and leads to a reduction in the computational effort for matrix coefficients and residuals evaluations numerical examples are shown to validate our approach accessing its efficiency and accuracy copyright 2007 john wiley sons ltd

trajectory statistical solutions and liouville type equations for evolution equations abstract results and applications abstract in this article we first prove from the viewpoint of infinite dynamical system sufficient conditions ensuring the existence of trajectory statistical solutions for autonomous evolution equations then we establish that the constructed trajectory statistical solutions possess invariant property and satisfy a liouville type equation moreover we reveal that the equation describing the invariant property of the trajectory statistical solutions is a particular situation of the liouville type equation finally we study the equations of three dimensional incompressible magneto micropolar fluids in detail and illustrate how to apply our abstract results to some concrete autonomous evolution equations

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

algebraic multilevel iteration method for lowest order raviart thomas space and applications an optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems that have their weak formulation in the space h div is presented the algorithm is developed for the discrete problem obtained by using the lowest order raviart thomas space the method is theoretically analyzed and supporting numerical examples are presented furthermore as a particular application the algorithm is used for the solution of the discrete minimization problem which arises in the functional type a posteriori error estimates for the discontinuous galerkin approximation of elliptic problems copyright 2011 john wiley sons ltd

a primal dual finite element method for first order transport problems abstract this article devises a new numerical method for first order transport problems by using the primal dual weak galerkin pd wg finite element method recently developed in scientific computing the pd wg method is based on a variational formulation of the modeling equation for which the differential operator is applied to the test function so that low regularity for the exact solution of the original equation is sufficient for computation the pd wg finite element method indeed yields a symmetric system involving both the original equation for the primal variable and its dual for the dual variable also known as lagrangian multiplier for the linear transport problem it is shown that the pd wg method offers numerical solutions that conserve mass locally on each element optimal order error estimates in various norms are derived for the numerical solutions arising from the pd wg method with weak regularity assumptions on the modelling equations a variety of numerical results are presented to demonstrate the accuracy and stability of the new method

partial regularity of suitable weak solutions to the system of the incompressible non newtonian fluids abstract in this paper we investigate the partial regularity of the generalized solutions which are called the suitable weak solutions to the modified navier stokes equations which describes the dynamics of the incompressible monopolar non newtonian fluids it is proved that the singular points are concentrated on a closed set whose 5 2p for p 5 2 dimensional hausdorff measure is zero and the solution is a regular one for p 5 2

invariant measures for multivalued semigroups abstract in this work we extend the concept of an invariant measure for a multivalued semigroup and when it has a global attractor we give different but equivalent definitions for such a measure as a consequence we can apply the birkhoff ergodic theorem to conclude that time averages converge almost everywhere to the spatial average

locking free reissner mindlin elements without reduced integration abstract in a recent paper of arnold et al d n arnold f brezzi l d marini a family of discontinuous galerkin finite elements for the reissner mindlin plate j sci comput 22 2005 25 45 the ideas of discontinuous galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the reissner mindlin plate problem by following their basic approach but making different choices of finite element spaces we develop and analyze other families of locking free finite elements that eliminate the need for the introduction of a reduction operator which has been a central feature of many locking free methods for k 2 all the methods use piecewise polynomials of degree k to approximate the transverse displacement and possibly subsets of piecewise polynomials of degree k 1 to approximate both the rotation and shear stress vectors the approximation spaces for the rotation and the shear stress are always identical the methods vary in the amount of interelement continuity required in terms of smallest number of degrees of freedom the simplest method approximates the transverse displacement with continuous piecewise quadratics and both the rotation and shear stress with rotated linear brezzi douglas marini elements

a posteriori error analysis for hybridizable discontinuous galerkin methods for second order elliptic problems we present a unified a posteriori error analysis for hybridizable discontinuous galerkin methods for second order elliptic partial differential equations the main feature of this analysis is that it reduces the task of establishing the reliability and efficiency of the estimator to the verification of three simple conditions this approach allows us not only to derive new estimates for hybridizable discontinuous galerkin methods but also to recover well known a posteriori error estimates for mixed methods and for the continuous galerkin method

a picone identity for half linear elliptic equations and its applications to oscillation theory abstract a picone identity is established for a class of half linear elliptic operators and sturmian comparison and oscillation results are derived on the basis of the picone identity wirtinger inequalities and riccati inequalities are also obtained as corollaries of the picone identity and a new class of half linear elliptic equations is introduced

h2 compact attractor for a non newtonian system in two dimensional unbounded domains abstract in this paper the authors study the long time behavior of a non newtonian system in two dimensional unbounded domains they prove the existence of h2 compact attractor for the system by showing the corresponding semigroup is asymptotically compact

partially penalized immersed finite element methods for elliptic interface problems this article presents new immersed finite element ife methods for solving the popular second order elliptic interface problems on structured cartesian meshes even if the involved interfaces have nontrivial geometries these ife methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in ife functions with the enhanced stability due to the added penalty not only can these ife methods be proven to have the optimal convergence rate in an energy norm provided that the exact solution has sufficient regularity but also numerical results indicate that their convergence rates in both the h 1 norm and the l 2 norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic ife methods in some situations trace inequalities are established for both linear and bilinear ife functions that are not only critical for the error analysis of these new ife methods but are also of a great potential to be useful in error analysis for other

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

global regularity for 2d magneto micropolar equations with only micro rotational velocity dissipation and magnetic diffusion abstract this paper studies the global regularity of classical solutions to 2d magneto micropolar fluid equations with only micro rotational velocity dissipation and magnetic diffusion here the micro rotational velocity dissipation and magnetic diffusion are given by and b making use of several combined quantities maximal regularity of heat operator and littlewood paley decomposition theory we establish a regularity criterion in terms of magnetic field for the case 1 and the global regularity for 1 the regularity criterion given here is also new even for the 2d magnetohydrodynamic equations in addition to prove these two main results as preparation we establish a new global a priori estimate for magnetic field namely b l 0 t l p r 2 with p 2 which also holds for the 2d magnetohydrodynamic equations as a particular case

abstract framework for the theory of statistical solutions abstract an abstract framework for the theory of statistical solutions is developed for general evolution equations extending the theory initially developed for the three dimensional incompressible navier stokes equations the motivation for this concept is to model the evolution of uncertainties on the initial conditions for systems which have global solutions that are not known to be unique both concepts of statistical solution in trajectory space and in phase space are given and the corresponding results of existence of statistical solution for the associated initial value problems are proved the wide applicability of the theory is illustrated with the very incompressible navier stokes equations a reaction diffusion equation and a nonlinear wave equation all displaying the property of global existence of weak solutions without a known result of global uniqueness

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

an adaptive stabilization strategy for enhanced strain methods in non linear elasticity this paper proposes and analyzes an adaptive stabilization strategy for enhanced strain es methods applied to quasistatic non linear elasticity problems the approach is formulated for any type of enhancements or material models and it is distinguished by the fact that the stabilization term is solution dependent the stabilization strategy is first constructed for general linearized elasticity problems and then extended to the non linear elastic regime via an incremental variational principle a heuristic choice of the stabilization parameters is proposed which in the numerical examples proved to provide stable approximations for a large range of deformations different problems and material models we also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable these are not advocated since they are generally larger than the ones based on heuristics and hence prone to deteriorate the locking free behavior of es methods numerical examples with two different non linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations finally the method is strongly based on earlier developments for discontinuous galerkin methods and hence throughout the paper we offer a perspective about the similarities between the two

adjoint consistency analysis of discontinuous galerkin discretizations this paper is concerned with the adjoint consistency of discontinuous galerkin dg discretizations adjoint consistency in addition to consistency is the key requirement for dg discretizations to be of optimal order in l 2 as well as measured in terms of target functionals we provide a general framework for analyzing the adjoint consistency of dg discretizations which is also useful for the derivation of adjoint consistent methods this analysis will be performed for the dg discretizations of the linear advection equation the interior penalty dg method for elliptic problems and the dg discretization of the compressible euler equations this framework is then used to derive an adjoint consistent dg discretization of the compressible navier stokes equations numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions

convergence of time averages of weak solutions of the three dimensional navier stokes equations using the concept of stationary statistical solution which generalizes the notion of invariant measure it is proved that in a suitable sense time averages of almost every leray hopf weak solution of the three dimensional incompressible navier stokes equations converge as the averaging time goes to infinity this system of equations is not known to be globally well posed and the above result answers a long standing problem extending to this system a classical result from ergodic theory it is also shown that from a measure theoretic point of view the stationary statistical solution obtained from a generalized limit of time averages is independent of the choice of the generalized limit finally any borel subset of the phase space with positive measure with respect to a stationary statistical solution is such that for almost all initial conditions in that borel set and for at least one leray hopf weak solution starting with that initial condition the corresponding orbit is recurrent to that borel subset and its mean sojourn time within that borel subset is strictly positive

a note on the asymptotic smoothing effect of solutions to a non newtonian system in 2 d unbounded domains abstract in this note the authors consider the regularity of the global attractor for a non newtonian system in two dimensional unbounded domains they prove that the l 2 compact attractor and the h 2 compact attractor of the system are the same and thus establish the asymptotic smoothing effect of solutions to this system

on finite element methods for heterogeneous elliptic problems dealing with variational formulations of second order elliptic problems with discontinuous coefficients we recall a single field minimization problem of an extended functional presented by bevilacqua et al bevilacqua l feijoo r a rojas l f 1974 a variational principle for the laplace operator with application in the torsion of composite rods international journal of solids structuring 10 1091 1102 which we associate with the basic idea supporting discontinuous galerkin finite element methods we review residual based stabilized mixed methods applied to darcy flow in homogeneous porous media and extend them to heterogeneous media with an interface of discontinuity for smooth interfaces the proposed formulations preserve the continuity of the flux and exactly imposes the constraint between the tangent components of darcy velocity on the interface convergence studies for a heterogeneous and anisotropic porous medium confirm the same rates of convergence predicted for homogeneous problem with smooth solutions

a hybrid discontinuous in space and time galerkin method for wave propagation problems summary medium frequency regime and multi scale wave propagation problems have been a subject of active research in computational acoustics recently new techniques have attempted to overcome the limitations of existing discretization methods that tend to suffer from dispersion one such technique the discontinuous enrichment method incorporates features of the governing partial differential equation in the approximation in particular the solutions of the homogeneous form of the equation here based on this concept and by extension of a conventional space time finite element method a hybrid discontinuous galerkin method dgm for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions is described the discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation in this contribution within each finite element the solutions in the form of polynomial waves are employed the continuity of these polynomial waves is weakly enforced through suitably chosen lagrange multipliers results for two dimensional and three dimensional problems in both low frequency and medium frequency regimes show that the proposed dgm outperforms the conventional space time finite element method copyright 2014 john wiley sons ltd

a two level enriched finite element method for a mixed problem the simplest pair of spaces is made inf sup stable for the mixed form of the darcy equation the key ingredient is to enhance the finite element spaces inside a petrov galerkin framework with functions satisfying element wise local darcy problems with right hand sides depending on the residuals over elements and edges the enriched method is symmetric locally mass conservative and keeps the degrees of freedom of the original interpolation spaces first we assume local enrichments exactly computed and we prove uniqueness and optimal error estimates in natural norms then a low cost two level finite element method is proposed to effectively obtain enhancing basis functions the approach lays on a two scale numerical analysis and shows that well posedness and optimality is kept despite the second level numerical approximation several numerical experiments validate the theoretical results and compares favourably in some cases our results with the classical raviart thomas element

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

weak pullback attractors of setvalued processes weak pullback attractors are defined for nonautonomous setvalued processes and their existence and upper semicontinuous convergence under perturbation is established unlike strong pullback attractors invariance and pullback attraction here are required only for at least one trajectory rather than all trajectories at each starting point the concept is useful in for example continuous time control systems and is related to that of viability

pullback attractors for a non autonomous incompressible non newtonian fluid this paper studies the pullback asymptotic behavior of solutions for a non autonomous incompressible non newtonian fluid in two dimensional 2d bounded domains we first prove the existence of pullback attractors av in space v has h2 regularity see notation in section 2 and ah in space h has l2 regularity for the cocycle corresponding to the solutions of the fluid then we verify the regularity of the pullback attractors by showing av ah which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data

low order discontinuous galerkin methods for second order elliptic problems we consider dg methods for second order scalar elliptic problems using piecewise affine approximation in two or three space dimensions we prove that both the symmetric and the nonsymmetric versions of the dg method have regular system matrices without penalization of the interelement solution jumps provided boundary conditions are imposed in a certain weak manner optimal convergence is proved for sufficiently regular meshes and data we then propose a dg method using piecewise affine functions enriched with quadratic bubbles using this space we prove optimal convergence in the energy norm for both a symmetric and nonsymmetric dg method without stabilization all of these proposed methods share the feature that they conserve mass locally independent of the penalty parameter

existence regularity and decay rate of solutions of non newtonian flow the existence and the regularity of young measure valued solutions to non newtonian flows are considered furthermore the uniqueness of solutions and their asymptotic behavior are given

an explicit discontinuous galerkin method for non linear solid dynamics formulation parallel implementation and scalability properties an explicit dynamics spatially discontinuous galerkin dg formulation for non linear solid dynamics is proposed and implemented for parallel computation dg methods have particular appeal in problems involving complex material response e g non local behavior and failure as even in the presence of discontinuities they provide a rigorous means of ensuring both consistency and stability in the proposed method these are guaranteed the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation the semi discrete system of ordinary differential equations is integrated in time using a conventional second order central difference explicit scheme a stability criterion for the time integration algorithm accounting for the influence of the dg discretization stability is derived for the equivalent linearized system this approach naturally lends itself to efficient parallel implementation the resulting dg computational framework is implemented in three dimensions via specialized interface elements the versatility robustness and scalability of the overall computational approach are all demonstrated in problems involving stress wave propagation and large plastic deformations copyright 2007 john wiley sons ltd

an a posteriori error analysis of an augmented discontinuous galerkin formulation for darcy flow in this paper we develop an a posteriori error analysis for an augmented discontinuous garlerkin formulation applied to the darcy flow more precisely we derive a reliable and efficient a posteriori error estimator which consists of residual terms finally we present several numerical experiments showing the robustness of the method and the theoretical properties of the estimator thus confirming the capability of the corresponding adaptive algorithms to localize the inner layers the singularities and or the large stress regions of the exact solution

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

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

a new c0 discontinuous galerkin method for kirchhoff plates abstract a general framework of constructing c 0 discontinuous galerkin cdg methods is developed for solving the kirchhoff plate bending problem following some ideas in castillo et al 2000 10 and cockburn 2003 12 the numerical traces are determined based on a discrete stability identity which lead to a class of stable cdg methods a stable cdg method called the lcdg method is particularly interesting in our study it can be viewed as an extension to fourth order problems of the ldg method studied in castillo et al 2000 10 and cockburn 2003 12 for this method optimal order error estimates in certain broken energy norm and h 1 norm are established some numerical results are reported confirming the theoretical convergence orders

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

multilevel preconditioners for discontinuous galerkin approximations of elliptic problems with jump coefficients in this article we develop and analyze two level and multi level methods for the family of interior penalty ip discontinuous galerkin dg discretizations of second order elliptic problems with rough coecients exhibiting large jumps across interfaces in the domain these methods are based on a decomposition of the dg nite element space that inherently hinges on the diusion coecient of the problem our analysis of the proposed preconditioners is presented for both symmetric and non symmetric ip schemes and we establish both robustness with respect to the jump in the coecient and near optimality with respect to the mesh size following the analysis we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods

an interior penalty stabilised incompressible discontinuous galerkin fourier solver for implicit large eddy simulations abstract we present an implicit large eddy simulation iles h p high order 2 unstructured discontinuous galerkin fourier solver with sliding meshes the solver extends the laminar version of ferrer and willden 2012 34 to enable the simulation of turbulent flows at moderately high reynolds numbers in the incompressible regime this solver allows accurate flow solutions of the laminar and turbulent 3d incompressible navier stokes equations on moving and static regions coupled through a high order sliding interface the spatial discretisation is provided by the symmetric interior penalty discontinuous galerkin ip dg method in the x y plane coupled with a purely spectral method that uses fourier series and allows efficient computation of spanwise periodic three dimensional flows since high order methods e g discontinuous galerkin and fourier are unable to provide enough numerical dissipation to enable under resolved high reynolds computations i e as necessary in the iles approach we adapt the laminar version of the solver to increase controllably the dissipation and enhance the stability in under resolved simulations the novel stabilisation relies on increasing the penalty parameter included in the dg interior penalty ip formulation the latter penalty term is included when discretising the linear viscous terms in the incompressible navier stokes equations these viscous penalty fluxes substitute the stabilising effect of non linear fluxes which has been the main trend in implicit les discontinuous galerkin approaches the ip dg penalty term provides energy dissipation which is controlled by the numerical jumps at element interfaces e g large in under resolved regions such as to stabilise under resolved high reynolds number flows this dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps thus providing an efficient stabilization mechanism for iles the ip dg stabilisation is complemented with a spectral vanishing viscosity svv method in the z direction to enhance stability in the continuous fourier space the coupling between the numerical viscosity in the dg plane and the svv damping provides an efficient approach to stabilise high order methods at moderately high reynolds numbers we validate the formulation for three turbulent flow cases a circular cylinder at re 3900 a static and pitch oscillating naca 0012 airfoil at re 10000 and finally a rotating vertical axis turbine at re 40000 with reynolds based on the circular diameter airfoil chord and turbine diameter respectively all our results compare favourably with published direct numerical simulations large eddy simulations or experimental data we conclude that the dg fourier high order solver with ip svv stabilisation proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery

regularity of trajectory attractor and upper semicontinuity of global attractor for a 2d non newtonian fluid abstract there are two results within this paper the one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non newtonian fluid on 2d bounded domains for which the solution to each initial value could be non unique the other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from m to r l l where m m 1 is an expanding sequence of simply connected bounded and smooth subdomains of such that m as m that is let a and a m be the global attractors of the fluid corresponding to and m respectively we establish that for any neighborhood o a of a the global attractor a m enters o a if m is large enough

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

numerical validation of a class of mixed discontinuous galerkin methods for darcy flow we present a thorough set of numerical experiments to confirm the robustness of several discontinuous galerkin methods proposed for the approximation of darcy law we perform two and three dimensional tests involving both low and high order approximations on cartesian grids and on triangular grids the tests we present confirm the theoretical estimates for symmetric approximations and characterize the behavior of non symmetric approximations where convergence rates that are one order better than predicted are observed for suitable choices of approximation spaces robustness and accuracy of all these methods are assessed for realistic numerical experiments where discontinuous permeability coefficients and bounded smooth permeability functions are used

pin wise homogenization for spn neutron transport approximation using the finite element method the neutron transport equation describes the distribution of neutrons inside a nuclear reactor core homogenization strategies have been used for decades to reduce the spatial and angular domain complexity of a nuclear reactor by replacing previously calculated heterogeneous subdomains by homogeneous ones and using a low order transport approximation to solve the new problem the generalized equivalence theory for homogenization looks for discontinuous solutions through the introduction of discontinuity factors at the boundaries of the homogenized subdomains in this work the generalized equivalence theory is extended to the simplified p n equations using the finite element method this extension proposes pin discontinuity factors instead of the usual assembly discontinuity factors and the use of the simplified spherical harmonics approximation rather than diffusion theory an interior penalty finite element method is used to discretize and solve the problem using discontinuity factors one dimensional numerical results show that the proposed pin discontinuity factors produce more accurate results than the usual assembly discontinuity factors the proposed pin discontinuity factors produce precise results for both pin and assembly averaged values without using advanced reconstruction methods also the homogenization methodology is verified against the calculation performed with reference discontinuity factors

asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3d globally modified navier stokes equations abstract we first prove the existence and regularity of the trajectory attractor for a three dimensional system of globally modified navier stokes equations then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space in our construction the trajectory statistical solution is an invariant borel probability measure which is supported by the trajectory attractor and is invariant under the action of the translation semigroup as a byproduct of the regularity of the trajectory attractor we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions

large time behavior to the system of incompressible non newtonian fluids in r2 abstract in this paper the authors study the large time behavior for the weak solutions to a class system of the incompressible non newtonian fluids in r 2 it is proved that the weak solutions decay in l 2 norm at 1 t 1 2 and the estimate for the decay rate is sharp in the sense that it coincides with the decay rate of a solution to the heat equation

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

global strong solutions to the 3d incompressible mhd equations with density dependent viscosity abstract this paper is concerned with the 3d incompressible mhd equations with density dependent viscosity in smooth bounded domains through time weighted a priori estimates the global existence of strong solutions is established under the assumption that the initial energy is suitably small this generalizes previous results for the 3d navier stokes equations in huang and wang 2015 and zhang 2015 which need u 0 l 2 to be small the initial vacuum is allowed

a space time discontinuous galerkin method for the solution of the wave equation in the time domain in recent years the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation this has led to the development of new concepts including the discontinuous enrichment method its basic principle is the incorporation of features of the governing partial differential equation in the approximation in this contribution this concept is adapted for the simulation of transient problems governed by the wave equation we present a space time discontinuous galerkin method with lagrange multipliers where the shape approximation in space and time is based on solutions of the homogeneous wave equation the use of hierarchical wave like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space time finite element methods copyright 2008 john wiley sons ltd

element boundary terms in reduced order models for flow problems domain decomposition and adaptive coarse mesh hyper reduction abstract in this paper we present a finite element based reduced order model and in particular we consider two aspects related to the introduction of inter element boundary terms in the formulation the first is a domain decomposition strategy in which the transmission conditions involve boundary terms to account for non matching meshes and discontinuous physical properties the second is a coarse mesh hyper reduction for which we propose an adaptive refinement driven by an a posteriori error estimator that contains element boundary terms as the finite element full order model the reduced order model is based on the variational multi scale framework with sub grid scales defined not only in the element interiors but also on the inter element boundaries we present some examples of application using the incompressible navier stokes equations and the boussinesq approximation

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

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

local error analysis of discontinuous galerkin methods for advection dominated elliptic linear quadratic optimal control problems this paper analyzes the local properties of the symmetric interior penalty upwind discontinuous galerkin sipg method for the numerical solution of optimal control problems governed by linear reaction advection diffusion equations with distributed controls the theoretical and numerical results presented in this paper show that for advection dominated problems the convergence properties of the sipg discretization can be superior to the convergence properties of stabilized finite element discretizations such as the streamline upwind petrov galerkin supg method for example we show that for a small diffusion parameter the sipg method is optimal in the interior of the domain this is in sharp contrast to supg discretizations for which it is known that the existence of boundary layers can pollute the numerical solution of optimal control problems everywhere even into domains where the solution is smooth and as a consequence in general reduces the convergence rates to only first order in order to prove

trajectory attractor and global attractor for a two dimensional incompressible non newtonian fluid the authors construct the trajectory attractor and global attractor for an autonomous two dimensional non newtonian fluid

picone identities for half linear elliptic operators with p x laplacians and applications to sturmian comparison theory picone identities are established for a class of half linear elliptic operators with p x laplacians and sturmian comparison theorems are obtained on the basis of the picone identities specializations to half linear partial or ordinary differential inequalities with p x laplacians are shown

long time behavior for the weakly damped driven long wave short wave resonance equations abstract in this paper we consider the cauchy problem of the long wave short wave resonance equations by making use of a strichartz type inequality for the solutions decomposing suitably the solution semigroup into a decay parts and a more regular parts and ruling out the vanishing and dichotomy of the solutions we prove the existence of the global attractor and the asymptotic smoothing effect of the solutions

picone identity for quasilinear elliptic equations with p x laplacians and sturmian comparison theory picone identities for quasilinear elliptic equations with p x laplacians are established and sturmian comparison theory is developed on the basis of picone identities as specializations various elliptic equations such as linear elliptic equations half linear elliptic equations with p laplacians half linear elliptic equations with p x laplacians pseudo p laplacian with variable power are discussed generalizations are also investigated

sturmian comparison and oscillation theorems for a class of half linear elliptic equations abstract sturmian comparison theorems are established for a class of half linear elliptic equations via the picone identity and oscillation results are derived on the basis of comparison results obtained

asymptotic smoothing effect of solutions to weakly dissipative klein gordon schrodinger equations abstract in this paper the authors consider the cauchy problem of weakly dissipative klein gordon schrodinger equations through yukawa coupling in r 3 making use of a strichartz type inequality and a suitable decomposition of the solution semigroup they prove the asymptotic smoothing effect of the solutions

global regularity to the 3d incompressible mhd equations abstract in this paper we prove the global regularity to the 3d nonhomogeneous incompressible magnetohydrodynamic equations let 0 m 0 and h 0 be the initial density momentum and magnetic field respectively we establish a unique strong solution on r 3 0 t for any 0 t under the assumption that the viscosity coefficient 0 is sufficiently large or m 0 2 0 l 1 2 h 0 l 2 2 or u 0 l 2 2 h 0 h 1 2 is small enough moreover if the given data are more regular and satisfy an additional compatibility condition for the existence of strong solution then we show that the strong solution is indeed a classical one moreover the weak strong uniqueness of solutions is also obtained which generalizes the result in 13 to the case of vacuum

a family of multiscale hybrid mixed finite element methods for the darcy equation with rough coefficients abstract we aim at proposing novel stable finite element methods for the mixed darcy equation with heterogeneous coefficients within a space splitting framework we start from the primal hybrid formulation of the elliptic model for the pressure localization of this infinite dimensional problem leads to element level boundary value problems which embed multiscale and high contrast features in a natural way with neumann boundary conditions driven by the lagrange multipliers such a procedure leads to methods involving the space of piecewise constants for the pressure together with a discretization of the fluxes choosing arbitrarily polynomial interpolations the lowest order raviart thomas element as well as some recent multiscale methods are recovered in addition the methods assure local mass conservation and can be interpreted as stabilized primal hybrid methods extensive numerical validation attests to the accuracy of the new methods on academic and more realistic problems with rough coefficients

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

the global attractor for the 2d navier stokes flow on some unbounded domains we extend previous results on the existence of the global attractor for the 2d navier stokes equations on some unbounded domains in the sense that the forcing term need not lie in any weighted space nor is the boundary of the domain required to be smooth the existence of the global attractor is obtained on arbitrary open sets such that the poincare inequality holds and for forces in the natural dual space v the proof is based on the energy equation and the concept of asymptotic compactness an estimate of the hausdorff and fractal dimensions of the attractor is also given

numerical verification of a non residual orthogonal term by term stabilized finite element formulation for incompressible convective flow problems abstract the modeling of turbulent flows is relevant in many engineering applications and is an active field of research on numerical methods convergence and stability of proposed formulations are crucial to predict transitional flows from laminar to turbulent flows in this work a recently developed stabilized finite element formulation is used as a powerful tool to describe such kind of problems the essential point of the formulation is the time dependent nature of the subscales and contrary to residual based formulations the introduction of two velocity subscale components the theoretical rate of convergence of the method is verified numerically using linear and quadratic equal order finite element discretizations to this end a standard convergence test of l 2 norm is presented where the computed solutions are compared when manufactured solutions are imposed at gauss point level moreover the hopf bifurcation is studied for two well known benchmark problems flow past a cylinder and the three dimensional lid driven cavity flow for the flow past a cylinder case the hopf bifurcation is verified using dynamic subscales and is assessed so that they do not disturb the solution in particular a dominant convective problem re 4000 is solved using both the quasistatic and dynamic versions of the method evaluating the performance of each one in the quality of the solution and in the cpu time needed to obtain a converged solution for the 3d lid driven cavity flow problem the hopf bifurcation is determined using two different boundary conditions analyzing their effect on the dynamics of the problem and on the thickness and shape of the boundary layer the final test case is the turbulent 3d lid driven cavity problem re 12000 where velocity profiles are compared with experimental les and dns reference solutions additionally pressure and velocity spectra are shown at certain representative points of the domain as well as phase diagrams correlation function graphs the poincare map and the lyapunov exponent typical mathematical tools used in dynamical system analysis from the results the method is robust and accurate for all the numerical tests both in viscous dominant problems as in dominant convective ones

analytical and variational numerical methods for unstable miscible displacement flows in porous media the miscible displacement of one fluid by another in a porous medium has received considerable attention in sub surface environmental and petroleum engineering applications when a fluid of higher mobility displaces another of lower mobility unstable patterns referred to as viscous fingering may arise their physical and mathematical study has been the object of numerous investigations over the past century the objective of this paper is to present a review of these contributions with particular emphasis on variational methods these algorithms are tailored to real field applications thanks to their advanced features handling of general complex geometries robustness in the presence of rough tensor coefficients low sensitivity to mesh orientation in advection dominated scenarios and provable convergence with fully unstructured grids this paper is dedicated to the memory of dr jim douglas jr for his seminal contributions to miscible displacement and variational numerical methods

efficient computable error bounds for discontinuous galerkin approximations of elliptic problems we present guaranteed and computable both sided error bounds for the discontinuous galerkin dg approximations of elliptic problems these estimates are derived in the full dg norm on purely functional grounds by the analysis of the respective differential problem and thus are applicable to any qualified dg approximation based on the triangle inequality the underlying approach has the following steps for a given dg approximation 1 computing a conforming approximation in the energy space using the oswald interpolation operator and 2 application of the existing functional a posteriori error estimates to the conforming approximation various numerical examples with varying difficulty in computing the error bounds from simple problems of polynomial type analytic solution to problems with analytic solution having sharp peaks or problems with jumps in the coefficients of the partial differential equation operator are presented which confirm the efficiency and the robustness of the estimates