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

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

moment lyapunov exponent and stability index for linear conservative system with small random perturbation asymptotic expansion series for the moment lyapunov exponent and stability index are constructed and justified for the two dimensional linear stochastic system close to a harmonic oscillator as an example a one degree of freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained

orthogonal bases for vertex mapped pyramids discontinuous galerkin dg methods discretized under the method of lines must handle the inverse of a block diagonal mass matrix at each time step efficient implementations of the dg method hinge upon inexpensive and low memory techniques for the inversion of each dense mass matrix block we propose an efficient time explicit dg method on meshes of pyramidal elements based on the construction of a seminodal high order basis which is orthogonal for a class of transformations of the reference pyramid despite the nonaffine nature of the mapping we give numerical results confirming both expected convergence rates and discuss efficiency of dg methods under such a basis

a high order discontinuous galerkin method for wave propagation through coupled elastic acoustic media we introduce a high order discontinuous galerkin dg scheme for the numerical solution of three dimensional 3d wave propagation problems in coupled elastic acoustic media a velocity strain formulation is used which allows for the solution of the acoustic and elastic wave equations within the same unified framework careful attention is directed at the derivation of a numerical flux that preserves high order accuracy in the presence of material discontinuities including elastic acoustic interfaces explicit expressions for the 3d upwind numerical flux derived as an exact solution for the relevant riemann problem are provided the method supports h non conforming meshes which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength as well as dynamic adaptivity to track solution features the use of high order elements controls numerical dispersion enabling propagation over many wave periods we prove consistency and stability of the proposed dg scheme to study the numerical accuracy and convergence of the proposed method we compare against analytical solutions for wave propagation problems with interfaces including rayleigh lamb scholte and stoneley waves as well as plane waves impinging on an elastic acoustic interface spectral rates of convergence are demonstrated for these problems which include a non conforming mesh case finally we present scalability results for a parallel implementation of the proposed high order dg scheme for large scale seismic wave propagation in a simplified earth model demonstrating high parallel efficiency for strong scaling to the full size of the jaguar cray xt5 supercomputer

moment lyapunov exponents of the stochastic parametrical hill s equation abstract the lyapunov exponent and moment lyapunov exponents of hill s equation with frequency and damping coefficient fluctuated by white noise stochastic process are investigated a perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises the results are applied to the study of the almost sure and the moment stability of the stationary solutions of the thin simply supported beam subjected to axial compressions and time varying damping which are small intensity stochastic excitations

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

non classical boundary conditions and dqm the dqm is applied to dynamic and stability analysis of beams with non classical boundary conditions the obtained results are compared with the exact frequencies and critical loads and the agreement is shown to be quite satisfactory for the entire parameter range

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

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

a new approach to the differential quadrature method for fourth order equations a generalized and more complete methodology for treating boundary conditions in the differential quadrature method dqm is presented this improved approach eliminates the deficiencies of the type grid arrangement which represents an approximation by applying the boundary conditions exactly two kinds of basis functions chebyshev and lagrange are used for concept demonstration it is found that the new approach cures most deficiencies of the current dqm 1997 by john wiley sons ltd

moment lyapunov exponents of a two dimensional system under bounded noise parametric excitation the moment lyapunov exponents of a two dimensional system under bounded noise parametric excitation are studied in this paper the method of regular perturbation is applied to obtain weak noise expansions of the moment lyapunov exponent lyapunov exponent and stability index in terms of the small fluctuation parameter

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

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

moment lyapunov exponents of a two dimensional system under combined harmonic and real noise excitations abstract the moment lyapunov exponents and the lyapunov exponents of a two dimensional system under combined excitations of harmonic and real noise which is modelled as an ornstein uhlenbeck process are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a regular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the real noise excitation on the parametric resonance due to the harmonic excitation is investigated

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

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

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

moment lyapunov exponents of a two dimensional system under both harmonic and white noise parametric excitations the moment lyapunov exponents and the lyapunov exponents of a 2d system under both harmonic and white noise excitations are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a singular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the white noise excitation on the parametric resonance due to the harmonic excitation is investigated

moment lyapunov exponents of a two dimensional system under real noise excitation abstract the moment lyapunov exponents of a two dimensional system under real noise excitation an ornstein uhlenbeck process are studied in this paper the method of regular perturbation is applied to obtain the weak noise expansions of the moment lyapunov exponent lyapunov exponent and stability index in terms of the small fluctuation parameter

large deviations and stochastic flows of diffeomorphisms previous results in the theory of large deviations for additive functionals of a diffusion process on a compact manifold m are extended and then applied to the analysis of the lyapunov exponents of a stochastic flow of diffeomorphisms of m an approximation argument relates these results to the behavior near the diagonal in m2 of the associated two point motion finally it is shown under appropriate non degeneracy conditions that the two point motion is ergodic on m2 if the top lyapunov exponent is positive

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

three approaches for the axial vibrations of bars on modified winkler soil with nonclassical boundary conditions abstract in this paper three approximate numerical approaches are compared for the title problem the first method is a variational one and it is known as an optimized rayleigh or rayleigh schmidt method as such it belongs to the so called energy approaches on the contrary the second method solves the differential equation of motion according to a recent quadrature procedure and is known as the differential quadrature method or dqm the last approach reduces the structure to an holonomic n degree of freedom mechanism the energies of which are easily written and the resulting equations of motion can be deduced by using the lagrange equations for discrete systems

a low storage curvilinear discontinuous galerkin method for wave problems the low storage curvilinear discontinuous galerkin lsc dg method reduces the storage requirements for solving symmetric and linear linear symmetric conservation laws in curvilinear domains when compared to the standard discontinuous galerkin method we perform a semidiscrete a priori convergence analysis of lsc dg and determine sufficient conditions on sequences of meshes to guarantee the same rate of convergence as the original upwind dg method on curvilinear domains computational results confirm the optimal order convergence on sample mesh sequences that satisfy the sufficiency conditions additional results show that the sufficient conditions for optimal order convergence of lsc dg may also be necessary conditions

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

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

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

the optimized rayleigh method andmathematicain vibrations and buckling problems abstract in this paper some vibration and buckling problems are solved by means of the optimized rayleigh and timoshenko quotients the use of the mathematica symbolic language produces closer approximations than the usual ones because two parameter quotients can be employed

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

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

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

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

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

vibration analysis of beams with non local foundations using the finite element method in this paper a non local viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method unlike local foundation models the reaction of the non local model is obtained as a weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure in the finite element analysis the interpolating shape functions of the element displacement field are identical to those of standard two node beam elements however for non local elasticity or damping nodes remote from the element do have an effect on the energy expressions and hence the damping and stiffness matrices the expressions of these direct and cross matrices for stiffness and damping may be obtained explicitly for some common spatial kernel functions alternatively numerical integration may be applied to obtain solutions numerical results for eigenvalues and associated eigenmodes of euler bernoulli beams are presented and compared where possible with results in literature using exact solutions and galerkin approximations the examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with non local viscoelastic foundations copyright 2007 john wiley sons ltd

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

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

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

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

moment lyapunov exponents and stochastic stability of a double beam system under compressive axial loading the lyapunov exponent and moment lyapunov exponents of two degrees of freedom linear systems subjected to white noise parametric excitation are investigated the method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises the lyapunov exponent and moment lyapunov exponents are important characteristics for determining the almost sure and moment stability of a stochastic dynamical system as an example we study the almost sure and moment stability of the double beam system under stochastic compressive axial loading the validity of the approximate results for moment lyapunov exponents is checked by the numerical monte carlo simulation method for this stochastic system

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

reduced storage nodal discontinuous galerkin methods on semi structured prismatic meshes abstract we present a high order time domain nodal discontinuous galerkin method for wave problems on hybrid meshes consisting of both wedge and tetrahedral elements we allow for vertically mapped wedges which can be deformed along the extruded coordinate and present a simple method for producing quasi uniform wedge meshes for layered domains we show that standard mass lumping techniques result in a loss of energy stability on meshes of vertically mapped wedges and propose an alternative which is both energy stable and efficient high order convergence is demonstrated and comparisons are made with existing low storage methods on wedges finally the computational performance of the method on graphics processing units is evaluated

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

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

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

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

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

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

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

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

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

mean square stability of a second order parametric linear system excited by a colored gaussian noise abstract the stochastic stability of a second order linear parametric oscillator whose stiffness is perturbed by a stationary zero mean colored gaussian process is investigated in this paper referring to the stability in the second response moments mean square stability the stochastic perturbation is the output of a linear filter excited by a stationary gaussian white noise stochastic process and the hypothesis of weak excitation is not formulated the first assumption allows using ito s stochastic differential calculus the moment equations of the response are written till the second order by means of ito s differential rule they form an infinite hierarchy that is the equations for the moments of order r contain moments of order larger than r in order to close the hierarchy the cumulant neglect closure method is applied herein the moment equation set is closed by neglecting the cumulants of third order nonlinear moment equations result they are linearized and the limit of stability is searched by studying the eigenvalues of the matrix of the coefficients of the linearized moment equations several numerical analyses are performed to evaluate the critical mean square value of the excitation when a system parameter is varied for the cases of both first and second order parametric excitation a first order excitation an ornstein uhlenbeck process does not cause stochastic resonance while a second order excitation does the first result was not known previously

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

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

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

a skew symmetric discontinuous galerkin spectral element discretization and its relation to sbp sat finite difference methods this paper shows that the discontinuous galerkin collocation spectral element method with gauss lobatto points dgsem gl satisfies the discrete summation by parts sbp property and can thus be classified as an sbp sat simultaneous approximation term scheme with a diagonal norm operator in the same way sbp sat finite difference schemes can be interpreted as discontinuous galerkin type methods with a corresponding weak formulation based on an inner product formulation common in the finite element community this relation allows the use of matrix vector notation common in the sbp sat finite difference community to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for dgsem gl and diagonal norm sbp sat based on this result a skew symmetric energy stable discretely conservative dgsem gl formulation applicable to general diagonal norm sbp sat schemes for the nonlinear burgers equation is constructed

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