unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

smoothed aggregation multigrid solvers for high order discontinuous galerkin methods for elliptic problems abstract we develop a smoothed aggregation based algebraic multigrid solver for high order discontinuous galerkin discretizations of the poisson problem algebraic multigrid is a popular and effective method for solving the sparse linear systems that arise from discretizing partial differential equations however high order discontinuous galerkin discretizations have proved challenging for algebraic multigrid the increasing condition number of the matrix and loss of locality in the matrix stencil as p increases in addition to the effect of weakly enforced dirichlet boundary conditions all contribute to the challenging algebraic setting we propose a smoothed aggregation approach that addresses these difficulties in particular the approach effectively coarsens degrees of freedom centered at the same spatial location as well as degrees of freedom at the domain boundary moreover the character of the near null space particularly at the domain boundary is captured by interpolation one classic prolongation smoothing step of weighted jacobi is also shown to be ineffective at high order and a more robust energy minimization approach is used along with block relaxation that more directly utilizes the block diagonal structure of the discontinuous galerkin discretization finally we conclude by examining numerical results in support our proposed method

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

on 2d elliptic discontinuous galerkin methods we discuss the discretization using discontinuous galerkin dg formulation of an elliptic poisson problem two commonly used dg schemes are investigated the original average flux proposed by bassi and rebay j comput phys 1997 131 267 and the local discontinuous galerkin ldg siam j numer anal 1998 35 2440 2463 scheme in this paper we expand on previous expositions discontinuous galerkin methods theory computation and applications springer berlin 2000 135 146 siam j sci comput 2002 24 2 524 547 int j numer meth engng 2003 58 2 1119 1148 by adopting a matrix based notation with a view to highlighting the steps required in a numerical implementation of the dg method through consideration of standard c0 type expansion bases as opposed to elementally orthogonal expansions with the matrix formulation we are able to apply static condensation techniques to improve efficiency of the direct solver when high order expansions are adopted the use of c0 type expansions also permits the direct enforcement of dirichlet boundary conditions through a lifting approach where the ldg flux does not require further stabilization in our construction we also adopt a formulation of the continuous dg fluxes that permits a more general interpretation of their numerical implementation in particular it allows us to determine the conditions under which the ldg method provides a near local stencil finally a study of the conditioning and the size of the null space of the matrix systems resulting from the dg discretization of the elliptic problem is undertaken copyright 2005 john wiley sons ltd

a hyperchaotic system from the rabinovich system this paper presents a new 4d hyperchaotic system which is constructed by a linear controller to the 3d rabinovich chaotic system some complex dynamical behaviors such as boundedness chaos and hyperchaos of the 4d autonomous system are investigated and analyzed a theoretical and numerical study indicates that chaos and hyperchaos are produced with the help of a lienard like oscillatory motion around a hypersaddle stationary point at the origin the corresponding bounded hyperchaotic and chaotic attractors are first numerically verified through investigating phase trajectories lyapunov exponents bifurcation path and poincare projections finally two complete mathematical characterizations for 4d hopf bifurcation are rigorously derived and studied

a high order elliptic pde based level set reinitialisation method using a discontinuous galerkin discretisation abstract in this paper an efficient high order accurate level set reinitialisation method is proposed based on the elliptic reinitialisation method basting and kuzmin 2013 1 which is discretised spatially using the discontinuous galerkin dg symmetric interior penalty method sipg in order to achieve this a number of improvements have been made to the elliptic reinitialisation method including reformulation of the underlying minimisation problem driving the solution adoption of a lagrange multiplier approach for enforcing a dirichlet boundary condition on the implicit level set interface and adoption of a narrow band approach numerical examples confirm the high order accuracy of the resultant method by demonstrating experimental orders of convergence congruent with optimal convergence rates for the sipg method that is h p 1 and h p in the l 2 and dg norms respectively furthermore the degree to which the level set function satisfies the eikonal equation improves proportionally to h p and the often ignored homogeneous dirichlet boundary condition on the interface is shown to be satisfied accurately with a rate of convergence of at least h 2 for all polynomial orders

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

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

a discontinuous hp finite element method for diffusion problems 1 d analysis abstract this paper presents the mathematical analysis of a new variant of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems not requiring auxiliary variables such as those used in mixed methods the focus of this study is on a class of linear second order boundary value problems for which we prove stability and a priori error estimates in both the finite and infinite dimensional spaces

a parabolic level set reinitialisation method using a discontinuous galerkin discretisation level set reinitialisation is a part of the level set methodology which allows one to generate at any point during level set evolution a level set function which is a signed distance function to its own zero isocontour whilst not in general a required condition maintaining the level set function as a signed distance function is often desirable as it removes a known source of numerical instability this paper presents a novel level set reinitialisation method based on the solution of a nonlinear parabolic pde the pde is discretised using a symmetric interior penalty discontinuous galerkin method in space and an implicit euler method in time also explored are explicit and semi implicit time discretisations however numerical experiments demonstrate that such methods suffer from severe time step restrictions leading to prohibitively large numbers of iterations required to achieve convergence the proposed method is shown to be high order accurate through a number of numerical examples more specifically the presented experimental orders of convergence align with the well established optimal convergence rates for the symmetric interior penalty method that is the error in the norm decreases proportionally to and the error in the dg norm decreases proportionally to

analysis of hdg methods for stokes flow in this paper we analyze a hybridizable discontinuous galerkin method for numerically solving the stokes equations the method uses polynomials of degree k for all the components of the approximate solution of the gradient velocity pressure formulation the novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method it renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges as a consequence we prove that the approximations of the velocity gradient the velocity and the pressure converge with the optimal order of convergence of k 1 in l 2 for any k 0 moreover taking advantage of the superconvergence properties of the velocity we introduce a new element by element postprocessing to obtain a new velocity approximation which is exactly divergence free h div conforming and converges with order k 2 for k 1 and with order 1 for k 0 numerical experiments are presented which validate the theoretical results

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity analytical estimates this is the second of two papers in which we motivate introduce and analyze a new type of strategy for the stabilization of discontinuous galerkin dg methods in nonlinear elasticity problems the foremost goal behind it is to enhance the robustness of the method without deteriorating the accuracy of the resulting solutions its distinctive property is that for nonlinear elastic problems the stabilization term is solution dependent and hence it is termed an adaptive stabilization mechanism the key contribution of this paper is the construction of a stabilization strategy for which the method is perfectly stable since the stabilization parameters can be explicitly computed as part of the numerical solution this is accomplished through the main result of this paper which consists of a theorem that provides lower bounds for the size of the stabilization parameters numerical examples confirm the guaranteed stability of the resulting method however they also show that the computed lower bounds overestimate the minimum amount of stabilization needed negatively affecting the approximation properties of the method these results underscore the fact that a better understanding of the stabilization mechanisms is needed in order to construct a method that is both robust and efficient

the cal p k 1 cal s k local discontinuous galerkin method for elliptic equations a local discontinuous galerkin ldg method using approximating spaces of order k 1 for the gradient and order k for the state variable with k ge 0 is presented for an elliptic boundary value problem a priori error estimates are derived and numerical results presented which demonstrate that this method is order k 1 for both the gradient and state variable in the l2 norm this improves upon earlier ldg methods which use equal order approximating spaces for both variables but lose an order of accuracy in the gradient approximation this approach is also the first convergent ldg method which allows for piecewise constant approximations for the state variable

a new hyperchaotic system from the lu system and its control this paper presents a 4d new hyperchaotic system which is constructed by a linear controller to a 3d lu system some complex dynamical behaviors such as hopf bifurcation chaos and hyperchaos of the simple 4d autonomous system are investigated and analyzed the corresponding hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories lyapunove exponents bifurcation path analysis of power spectrum and poincare projections furthermore the design is illustrated with both simulations and experiments finally the control problem of a new hyperchaotic system is investigated using negative feedback control ordinary feedback control dislocated feedback control and speed feedback control are used to suppress hyperchaos to an unstable equilibrium numerical simulations are presented to demonstrate the effectiveness of the proposed controllers

numerical simulation on hyperbolic diffusion equations using modified cubic b spline differential quadrature methods in this article a modified cubic b spline differential quadrature method mcb dqm is proposed to solve a hyperbolic diffusion problem in which flow motion is affected by both convection and diffusion one dimensional hyperbolic non homogeneous heat wave and telegraph equations are also considered along with two dimensional hyperbolic diffusion problem the method reduces the hyperbolic problem into a system of nonlinear ordinary differential equations the system is then solved by the optimal four stage three order strong stability preserving time stepping runge kutta ssp rk43 scheme the reliability and efficiency of the method has been tested on seven examples the stability of the method is also discussed and found to be unconditionally stable

a local discontinuous galerkin method for nonlinear diffusion problems with mixed boundary conditions in this paper we present and analyze a local discontinuous galerkin method for a class of nonlinear diffusion problems in polygonal regions of r 2 our analysis follows known approaches previously applied to linear problems and considers convex and nonconvex domains we provide solvability and stability of the discrete scheme for several polynomial approximations and we derive a priori error estimates in the energy and l2 norms numerical experiments illustrating these results are also provided

the direct coupling of local discontinuous galerkin and natural boundary element method for nonlinear interface problem in r 3 in this article we use the direct coupling of local discontinuous galerkin ldg and natural boundary element method nbem to solve a class of three dimensional interface problem which involves a nonlinear problem in a bounded domain and a poisson equation in an unbounded domain a spherical surface as an artificial boundary is introduced the coupled discrete primal formulation on a bounded domain is obtained the well posedness of the primal formulation is verified the optimal error order with respect to energy norm is given numerical examples are presented to demonstrate the optimal convergent rates

a discontinuous galerkin method for a hyperbolic model for convection diffusion problems in cfd this paper proposes a hyperbolic model for convection diffusion transport problems in computational fluid dynamics cfd the hyperbolic model is based on the so called cattaneo s law this is a time dependent generalization of fick s and fourier s laws that was originally proposed to solve pure diffusive heat transfer problems we show that the proposed model avoids the infinite speed paradox that is inherent in the standard parabolic model a high order upwind discontinuous galerkin dg method is developed and applied to classic convection dominated test problems the quality of the numerical results is remarkable since the discontinuities are very well captured without the appearance of spurious oscillations these results are compared with those obtained by using the standard parabolic model and the local dg ldg method and with those given by the parabolic model and the bassi rebay scheme finally the applicability of the proposed methodology is demonstrated by solving a practical case in engineering we simulate the evolution of pollutant being spilled in the harbour of a coruna northwest of spain eu

a multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method proliferation of degrees of freedom has plagued discontinuous galerkin methodology from its inception over 30 years ago this paper develops a new computational formulation that combines the advantages of discontinuous galerkin methods with the data structure of their continuous galerkin counterparts the new method uses local element wise problems to project a continuous finite element space into a given discontinuous space and then applies a discontinuous galerkin formulation the projection leads to parameterization of the discontinuous degrees of freedom by their continuous counterparts and has a variational multiscale interpretation this significantly reduces the computational burden and at the same time little or no degradation of the solution occurs in fact the new method produces improved solutions compared with the traditional discontinuous galerkin method in some situations

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

inexact hierarchical scale separation a two scale approach for linear systems from discontinuous galerkin discretizations abstract hierarchical scale separation hss is an iterative two scale approximation method for large sparse systems of linear equations arising from discontinuous galerkin dg discretizations hss splits the linear system into a coarse scale system of reduced size corresponding to the local mean values of the solution and a set of decoupled local fine scale systems corresponding to the higher order solution components this scheme then alternates between coarse scale and fine scale system solves until both components converge the motivation of hss is to promote parallelism by decoupling the fine scale systems and to reduce the communication overhead from classical linear solvers by only applying them to the coarse scale system we propose a modified hss scheme inexact hss ihss that exploits the highly parallel fine scale solver more extensively and only approximates the coarse scale solution in every iteration thus resulting in a significant speedup the tolerance of the coarse scale solver is adapted in every ihss cycle controlled by the residual norm of the fine scale system anderson acceleration is employed in the repeated solving of the fine scale system to stabilize the scheme we investigate the applicability of ihss to systems stemming from the nonsymmetric interior penalty dg discretization of the cahn hilliard equation discuss its hybrid parallel implementation for large scale simulations and compare the performance of a widely used iterative solver with and without ihss

spectral hp discontinuous galerkin methods for modelling 2d boussinesq equations we present spectral hp discontinuous galerkin methods for modelling weakly nonlinear and dispersive water waves described by a set of depth integrated boussinesq equations on unstructured triangular meshes when solving the equations two different formulations are considered directly solving the coupled momentum equations and the scalar method in which a wave continuity equation is solved as an intermediate step we demonstrate that the approaches are fully equivalent and give identical results in terms of accuracy convergence and restriction on the time step however the scalar method is shown to be more cpu efficient for high order expansions in addition to requiring less storage

error analysis of a modified discontinuous galerkin recovery scheme for diffusion problems abstract a theoretical error analysis using standard sobolev space energy arguments is furnished for a class of discontinuous galerkin dg schemes that are modified versions of one of those introduced by van leer and nomura these schemes which use discontinuous piecewise polynomials of degree q are applied to a family of one dimensional elliptic boundary value problems the modifications to the original method include definition of a recovery flux function via a symmetric l 2 projection and the addition of a penalty or stabilization term the method is found to have a convergence rate of o h q for the approximation of the first derivative and o h q 1 for the solution computational results for the original and modified dg recovery schemes are provided contrasting them as far as complexity and cost numerical examples are given which exhibit sub optimal convergence rates when the stabilization terms are omitted

scalable two level preconditioning and deflation based on a piecewise constant subspace for sip dg systems for diffusion problems this paper is focused on the preconditioned conjugate gradient cg method for linear systems resulting from symmetric interior penalty discontinuous galerkin sipg discretizations for stationary diffusion problems in particular it concerns two level preconditioning strategies where the coarse space is based on piecewise constant dg basis functions in this paper we show that both the two level preconditioner and the corresponding bnn or adef2 deflation variant yield scalable convergence of the cg method independent of the mesh element diameter these theoretical results are illustrated by numerical experiments we consider a solver for the discontinuous galerkin method we use the symmetric interior penalty variant the deflated solver appears to be scalable our solver is independent of the jump in the coefficients

the nonoccurrence of the lavrentiev phenomenon for a class of variational functionals we prove that the lavrentiev phenomenon does not occur for the functional int omega l nabla u x g u dx on u u 0 in w 1 1 0 omega where l and g are convex functions subject to additional regularity assumptions and where partial omega and u 0 are smooth

a hyperchaos generated from lorenz system abstract this paper presents a four dimension hyperchaotic lorenz system obtained by adding a nonlinear controller to lorenz chaotic system the hyperchaotic lorenz system is studied by bifurcation diagram lyapunov exponents spectrum and phase diagram numerical simulations show that the new system s behavior can be convergent divergent periodic chaotic and hyperchaotic when the parameter varies

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

superconvergent discontinuous galerkin methods for second order elliptic problems we identify discontinuous galerkin methods for second order elliptic problems in several space dimensions having superconvergence properties similar to those of the raviart thomas and the brezzi douglas marini mixed methods these methods use polynomials of degree k 0 for both the potential as well as the flux we show that the approximate flux converges in l2 with the optimal order of k 1 and that the approximate potential and its numerical trace superconverge in l2 like norms to suitably chosen projections of the potential with order k 2 we also apply element by element postprocessing of the approximate solution to obtain new approximations of the flux and the potential the new approximate flux is proven to have normal components continuous across inter element boundaries to converge in l2 with order k 1 and to have a divergence converging in l2 also with order k 1 the new approximate potential is proven to converge with order k 2 in l2 numerical experiments validating these theoretical results are presented

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

adaptive spacetime discontinuous galerkin method for hyperbolic advection diffusion with a non negativity constraint summary applications where the diffusive and advective time scales are of similar order give rise to advection diffusion phenomena that are inconsistent with the predictions of parabolic fickian diffusion models non fickian diffusion relations can capture these phenomena and remedy the paradox of infinite propagation speeds in fickian models in this work we implement a modified frame invariant form of cattaneo s hyperbolic diffusion relation within a spacetime discontinuous galerkin advection diffusion model an h adaptive spacetime meshing procedure supports an asynchronous patch by patch solution procedure with linear computational complexity in the number of spacetime elements this localized solver enables the selective application of optimization algorithms in only those patches that require inequality constraints to ensure a non negative concentration solution in contrast to some previous methods we do not modify the numerical fluxes to enforce non negative concentrations thus the element wise conservation properties that are intrinsic to discontinuous galerkin models are defined with respect to physically meaningful riemann fluxes on the element boundaries we present numerical examples that demonstrate the effectiveness of the proposed model and we explore the distinct features of hyperbolic advection diffusion response in subcritical and supercritical flows copyright 2015 john wiley sons ltd

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

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three 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

on a degenerate problem in the calculus of variations we establish the uniqueness of the solutions for a degenerate scalar problem in the multiple integrals calculus of variations the proof requires as a preliminary step the study of the regularity properties of the solutions and of their level sets we exploit the uniqueness and the regularity results to explore some of their qualitative properties in particular we emphasize the link between the supports of the solutions and the cheeger problem

a finite element formulation for a convection diffusion equation based on cattaneo s law a number of contributions have been made during the last decades to model pure diusive transport problems by using the so called hyperbolic diusion equa on problems based on a cattaneo type law the finite element solution of the proposed equations is addressed the performance of the algorithm is verified by solving some 2d test cases some interesting features of the proposed model can be observed from these examples and we conclude that the proposal is a feasible alternative to the parabolic model for real engineering simulations

an analysis of the embedded discontinuous galerkin method for second order elliptic problems the embedded discontinuous galerkin methods are obtained from hybridizable discontinuous galerkin methods by a simple change of the space of the hybrid unknown in this paper we consider embedded methods for second order elliptic problems obtained from hybridizable discontinuous methods by changing the space of the hybrid unknown from discontinuous to continuous functions this change results in a significantly smaller stiffness matrix whose size and sparsity structure coincides with those of the stiffness matrix of the statically condensed continuous galerkin method it is shown that this computational advantage has to be balanced against the fact that the approximate solutions for the scalar variable and its flux lose each a full order of convergence indeed we prove that if polynomials of degree k ge1 are used for the original hybridizable discontinuous galerkin method its approximations to the scalar variable and its flux converge with order k 2 and k 1 respectively whereas those of the corresponding embedded discontinuous galerkin method converge with orders k 1 and k respectively only we also provide numerical results comparing the relative efficiency of the methods

conservative super convergent and hybrid discontinuous galerkin methods applied to nonlinear schrodinger equations abstract using a unified framework the formulation of a super convergent discontinuous galerkin sdg method and a hybridized discontinuous galerkin hdg version both applied to a general nonlinear schrodinger equation is presented conservation of the mass and the energy is studied theoretically for the semi discrete formulation and for the fully discrete method using the modified crank nicolson time scheme conservation of both quantities is numerically validated on two dimensional problems and high order approximations a numerical study of convergence illustrates the advantages of the new formulations over the traditional local discontinuous galerkin ldg method numerical experiments show that the approximation of the initial discrete energy converges with order 2 k 1 which is better than that obtained by the standard continuous finite element which is only of order 2k when polynomials of degree k are used

uniqueness and comparison results for functionals depending on nabla u and on u we introduce a family of solutions to a variational problem and we prove uniqueness and comparison results

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

a hyperchaotic system from a chaotic system with one saddle and two stable node foci this paper presents a 4d new hyperchaotic system which is constructed by a linear controller to a 3d new chaotic system with one saddle and two stable node foci some complex dynamical behaviors such as ultimate boundedness chaos and hyperchaos of the simple 4d autonomous system are investigated and analyzed the corresponding bounded hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories lyapunove exponents bifurcation path analysis of power spectrum and poincare projections finally two complete mathematical characterizations for 4d hopf bifurcation are rigorous derived and studied

short note an explicit expression for the penalty parameter of the interior penalty method we derive an explicit expression for the penalty parameter of the interior penalty method for elliptic problems the expression yields a coercive bilinear form and is valid for general meshes comprising of geometrically nonconforming simplical elements

norm preconditioners for discontinuous galerkin hp finite element methods we consider a norm preconditioning approach for the solution of discontinuous galerkin finite element discretizations of second order partial differential equations with a nonnegative characteristic form our solution method is a norm preconditioned three term gmres routine we find that for symmetric positive definite diffusivity tensors the convergence of our solver is independent of discretization while for the semidefinite case both theory and experiment indicate dependence on both h and p numerical results are included to illustrate performance on several test cases

optimal trace inequality constants for interior penalty discontinuous galerkin discretisations of elliptic operators using arbitrary elements with non constant jacobians abstract in this paper a new method to numerically calculate the trace inequality constants which arise in the calculation of penalty parameters for interior penalty discretisations of elliptic operators is presented these constants are provably optimal for the inequality of interest as their calculation is based on the solution of a generalised eigenvalue problem involving the volumetric and face stiffness matrices the method is applicable to any element type for which these matrices can be calculated including standard finite elements and the non uniform rational b splines of isogeometric analysis in particular the presented method does not require the jacobian of the element to be constant and so can be applied to a much wider variety of element shapes than are currently available in the literature numerical results are presented for a variety of finite element and isogeometric cases when the jacobian is constant it is demonstrated that the new method produces lower penalty parameters than existing methods in the literature in all cases which translates directly into savings in the solution time of the resulting linear system when the jacobian is not constant it is shown that the naive application of existing approaches can result in penalty parameters that do not guarantee coercivity of the bilinear form and by extension the stability of the solution the method of manufactured solutions is applied to a model reaction diffusion equation with a range of parameters and it is found that using penalty parameters based on the new trace inequality constants result in better conditioned linear systems which can be solved approximately 11 faster than those produced by the methods from the literature

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

penalty method with p1 p1 finite element approximation for the stokes equations under the slip boundary condition we consider the p1 p1 or p1b p1 finite element approximations to the stokes equations in a bounded smooth domain subject to the slip boundary condition a penalty method is applied to address the essential boundary condition u cdot n g on partial omega which avoids a variational crime and simultaneously facilitates the numerical implementation we give o h 1 2 epsilon 1 2 h epsilon 1 2 error estimate for velocity and pressure in the energy norm where h and epsilon denote the discretization parameter and the penalty parameter respectively in the two dimensional case it is improved to o h epsilon 1 2 h 2 epsilon 1 2 by applying reduced order numerical integration to the penalty term the theoretical results are confirmed by numerical experiments

multigrid algorithms for symmetric discontinuous galerkin methods on graded meshes we study a class of symmetric discontinuous galerkin methods on graded meshes optimal order error estimates are derived in both the energy norm and the l 2 norm and we establish the uniform convergence of v cycle f cycle and w cycle multigrid algorithms for the resulting discrete problems numerical results that confirm the theoretical results are also presented

a superconvergence result for discontinuous galerkin methods applied to elliptic problems this paper presents a theoretical and numerical study of a class of discontinuous galerkin methods that shows the approximation of the gradient superconverges at the zeros of the legendre polynomials on a model 1d elliptic problem numerical experiments validate the theoretical results

a numerical study of a semi algebraic multilevel preconditioner for the local discontinuous galerkin method in this work we consider the local discontinuous galerkin ldg method applied to second order elliptic problems arising in the modeling of single phase flows in porous media it has been recently proven that the spectral condition number of the stiffness matrix exhibits an asymptotic behavior of o h 2 on structured and unstructured meshes where h is the mesh size thus efficient preconditioners are mandatory we present a semi algebraic multilevel preconditioner for the ldg method using local lagrange type interpolatory basis functions we show numerically that its performance does not degrade or at least the number of iterations increases very slowly as the number of unknowns augments the preconditioner is tested on problems with high jumps in the coefficients which is the typical scenario of problems arising in porous media

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

an alternating direction implicit method for a second order hyperbolic diffusion equation with convection a numerical method is presented to solve a two dimensional hyperbolic diffusion problem where is assumed that both convection and diffusion are responsible for flow motion since direct solutions based on implicit schemes for multidimensional problems are computationally inefficient we apply an alternating direction method which is second order accurate in time and space the stability of the alternating direction method is analyzed using the energy method numerical results are presented to illustrate the performance in different cases

a priori error estimates of a combined mixed finite element and local discontinuous galerkin method for an incompressible miscible displacement problem abstract a numerical approximation for a kind of incompressible miscible displacement problems in high dimension in porous media is studied mixed finite element method is applied to the flow equation and the transport one is solved by the local discontinuous galerkin method ldg based on interpolation projection properties and the induction hypothesis a priori hp error estimates are obtained numerical results are presented which verify the theoretical results

necessary conditions for solutions to variational problems we prove necessary conditions for a solution u to the problem of minimizing int omega f nabla v x g x v x dx in the form of a pontryagin maximum principle for f convex and satisfying a growth assumption but without assuming differentiability

analysis of schwarz methods for a hybridizable discontinuous galerkin discretization schwarz methods are attractive parallel solvers for large scale linear systems obtained when partial differential equations are discretized for hybridizable discontinuous galerkin hdg methods this is a relatively new field of research because hdg methods impose continuity across elements using a robin condition while classical schwarz solvers use dirichlet transmission conditions robin conditions are used in optimized schwarz methods to get faster convergence compared to classical schwarz methods and this even without overlap when the robin parameter is well chosen we present in this paper a rigorous convergence analysis of schwarz methods for the concrete case of the hybridizable interior penalty iph method we show that the penalization parameter needed for convergence of iph leads to slow convergence of the classical additive schwarz method and we propose a modified solver which leads to much faster convergence our analysis is entirely at the discrete level and thus holds for arbitrary in

superconvergent discontinuous galerkin methods for nonlinear elliptic equations based on the analysis of cockburn et al math comp 78 2009 pp 1 24 for a selfadjoint linear elliptic equation we rst discuss su perconvergence results for nonselfadjoint linear elliptic problems using discon tinuous galerkin methods further we have extended our analysis to derive superconvergence results for quasilinear elliptic problems when piecewise polynomials of degree k 1 are used to approximate both the potential as well as the ux it is shown in this article that the error estimate for the discrete ux in l 2 norm is of order k 1 further based on solving a discrete linear elliptic problem at each element a suitable postprocessing of the discrete potential is developed and then it is proved that the resulting post processed potential converges with order of convergence k 2 in l 2 norm these results conrm superconvergent results for linear elliptic problems

a high order discontinuous galerkin method for the unsteady incompressible navier stokes equations we present a high order discontinuous galerkin discretization of the unsteady incompressible navier stokes equations in convection dominated flows using triangular and tetrahedral meshes the scheme is based on a semi explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the stokes operator the nonlinear term is discretized in divergence form by using the local lax friedrichs fluxes thus local conservativity is inherent spatial discretization of the stokes operator has employed both equal order p k p k and mixed order p k p k 1 velocity and pressure approximations a second order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic helmholtz equation for each component of the velocity and a consistent poisson equation for the pressure the consistent poisson operator is replaced by an equivalent in stability and convergence operator namely that arising from the interior penalty discretization of the standard poisson operator with appropriate boundary conditions this yields a simpler and more efficient method characterized by a compact stencil size we show the temporal and spatial behavior of the method by solving some popular benchmarking tests for an unsteady stokes problem second order temporal convergence is obtained while for the taylor vortex test problem on both semi structured and fully unstructured triangular meshes spectral convergence with respect to the polynomial degree k is obtained by studying the orr sommerfeld stability problem we demonstrate that the p k p k method yields a stable solution while the p k p k 1 formulation leads to unphysical instability the good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder we conclude that the proposed discontinuous galerkin method with the p k p k formulation is an efficient scheme for accurate and stable solution of the unsteady navier stokes equations in convection dominated flows

on the global dynamics of the rabinovich system in this paper by using the poincare compactification in we make a global analysis of the rabinovich system with and we give the complete description of its dynamics on the sphere at infinity for ten sets of the parameter values the system has either first integrals or invariants for these ten sets we provide the global phase portrait of the rabinovich system in the poincare ball i e in the compactification of with the sphere of the infinity we prove that for convenient values of the parameters the system has two families of singularly degenerate heteroclinic cycles then changing slightly the parameters we numerically found a four wings butterfly shaped strange attractor

hp version discontinuous galerkin methods for hyperbolic conservation laws abstract the development of hp version discontinuous galerkin methods for hyperbolic conservation laws is presented in this work a priori error estimates are derived for a model class of linear hyperbolic conservation laws these estimates are obtained using a new mesh dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation the results generalize and extend previous results on mesh dependent norms to hp version discontinuous galerkin methods a posteriori error estimates which provide bounds on the actual error are also developed in this work numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution

the compact discontinuous galerkin cdg method for elliptic problems we present a compact discontinuous galerkin cdg method for an elliptic model problem the problem is first cast as a system of first order equations by introducing the gradient of the primal unknown or flux as an additional variable a standard discontinuous galerkin dg method is then applied to the resulting system of equations the numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level thus resulting in a global system that involves only the original unknown variable the proposed method is closely related to the local discontinuous galerkin ldg method b cockburn and c w shu siam j numer anal 35 1998 pp 2440 2463 but unlike the ldg method the sparsity pattern of the cdg method involves only nearest neighbors also unlike the ldg method the cdg method works without stabilization for an arbitrary orientation of the element interfaces the computation of the numerical interface fluxes for the cdg method is slightly more involved than for the ldg method but this additional complication is clearly offset by increased compactness and flexibility compared to the br2 f bassi and s rebay j comput phys 131 1997 pp 267 279 and ip j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 lecture notes in phys 58 springer berlin 1976 pp 207 216 methods which are known to be compact the present method produces fewer nonzero elements in the matrix and is computationally more efficient

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100