convergence studies on iterative algorithms for image reconstruction we introduce a general iterative scheme for image reconstruction based on landweber s method in our configuration a sequential block iterative seqbi version can be readily formulated from a simultaneous block iterative simbi version and vice versa this provides a mechanism to derive new algorithms from known ones it is shown that some widely used iterative algorithms such as the algebraic reconstruction technique art simultaneous art sart cimmino s and the recently designed diagonal weighting and component averaging algorithms are special examples of the general scheme we prove convergence of the general scheme under conditions more general than assumed in earlier studies for its seqbi and simbi versions in the consistent and inconsistent cases respectively our results suggest automatic relaxation strategies for the seqbi and simbi versions and characterize the dependence of the limit image on the initial guess it is found that in all cases the limit is the sum of the minimum norm solution of a weighted least squares problem and an oblique projection of the initial image onto the null space of the system matrix

compact and stable discontinuous galerkin methods for convection diffusion problems we present a new scheme the compact discontinuous galerkin 2 cdg2 method for solving nonlinear convection diffusion problems together with a detailed comparison to other well accepted dg methods the new cdg2 method is similar to the cdg method that was recently introduced in the work of perraire and persson for elliptic problems one main feature of the cdg2 method is the compactness of the stencil which includes only neighboring elements even for higher order approximation theoretical results showing coercivity and stability of cdg2 and cdg for the poisson and the heat equation are given providing computable bounds on any free parameters in the scheme in numerical tests for an elliptic problem a scalar convection diffusion equation and for the compressible navier stokes equations we demonstrate that the cdg2 method slightly outperforms similar methods in terms of l 2 accuracy and cpu time

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

iterative image reconstruction algorithms based on cross entropy minimization the related problems of minimizing the functionals f x alpha kl y px 1 alpha kl p x and g x alpha kl px y 1 alpha kl x p respectively over the set of vectors x or 0 are considered kl a b is the cross entropy or kullback leibler distance between two nonnegative vectors a and b iterative algorithms for minimizing both functionals using the method of alternating projections are derived a simultaneous version of the multiplicative algebraic reconstruction technique mart algorithm called smart is introduced and its convergence is proved

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

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

block iterative methods for image reconstruction from projections the simultaneous mart algorithm smart and the expectation maximization method for likelihood maximization emml are extended to block iterative versions bi smart and bi emml that converge to a solution in the feasible case for any choice of subsets the bi emml reduces to the ordered subset emml of hudson et al 1992 1994 when their subset balanced property holds

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

insights on the coercivity of the esfr methods for elliptic problems abstract the flux reconstruction approach is a recent high order method which has been introduced for unsteady problems initial energy stability has been conducted for the advection problem leading to the well know energy stable flux reconstruction esfr scheme using the esfr scheme the energy stability proof has been extended for the advection diffusion using the local discontinuous galerkin ldg numerical flux recently stability conditions were derived for the compact interior penalty ip and bassi rebay ii br2 numerical fluxes here we apply esfr schemes to elliptic problems and derive the associated bilinear form for the poisson equation we show that for the compact ip and br2 numerical fluxes the bilinear form is independent of the auxiliary correction function finally we provide some insights on the coercivity of the esfr scheme

a row action alternative to the em algorithm for maximizing likelihood in emission tomography the maximum likelihood ml approach to estimating the radioactive distribution in the body cross section has become very popular among researchers in emission computed tomography ect since it has been shown to provide very good images compared to those produced with the conventional filtered backprojection fbp algorithm the expectation maximization em algorithm is an often used iterative approach for maximizing the poisson likelihood in ect because of its attractive theoretical and practical properties its major disadvantage is that due to its slow rate of convergence a large amount of computation is often required to achieve an acceptable image here the authors present a row action maximum likelihood algorithm ramla as an alternative to the em algorithm for maximizing the poisson likelihood in ect the authors deduce the convergence properties of this algorithm and demonstrate by way of computer simulations that the early iterates of ramla increase the poisson likelihood in ect at an order of magnitude faster that the standard em algorithm specifically the authors show that from the point of view of measuring total radionuclide uptake in simulated brain phantoms iterations 1 2 3 and 4 of ramla perform at least as well as iterations 45 60 70 and 80 respectively of em moreover the authors show that iterations 1 2 3 and 4 of ramla achieve comparable likelihood values as iterations 45 60 70 and 80 respectively of em the authors also present a modified version of a recent fast ordered subsets em os em algorithm and show that ramla is a special case of this modified os em furthermore the authors show that their modification converges to a ml solution whereas the standard os em does not

high resolution viscous fingering simulation in miscible displacement using a p adaptive discontinuous galerkin method with algebraic multigrid preconditioner abstract high resolution simulation of viscous fingering can offer an accurate and detailed prediction for subsurface engineering processes involving fingering phenomena the fully implicit discontinuous galerkin dg method has been shown to be an accurate and stable method to model viscous fingering with high peclet number and mobility ratio in this paper we present two techniques to speedup large scale simulations of this kind the first technique relies on a simple p adaptive scheme in which high order basis functions are employed only in elements near the finger fronts where the concentration has a sharp change as a result the number of degrees of freedom is significantly reduced and the simulation yields almost identical results to the more expensive simulation with uniform high order elements throughout the mesh the second technique for speedup involves improving the solver efficiency we present an algebraic multigrid amg preconditioner which allows the dg matrix to leverage the robust amg preconditioner designed for the continuous galerkin cg finite element method the resulting preconditioner works effectively for fixed order dg as well as p adaptive dg problems with the improvements provided by the p adaptivity and amg preconditioning we can perform high resolution three dimensional viscous fingering simulations required for miscible displacement with high peclet number and mobility ratio in greater detail than before for well injection problems

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

time implicit high order discontinuous galerkin method with reduced evaluation cost an efficient and robust time integration procedure for a high order discontinuous galerkin method is introduced for solving nonlinear second order partial differential equations the time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term the procedure uses an iterative algorithm with reduced evaluation cost the size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between low order and high order degrees of freedom numerical examples are presented for the nonlinear convection diffusion equation in one and two dimensions including steady and unsteady flow problems the performance of the present method is investigated in terms of cpu time and compared to a fully implicit method a von neumann stability analysis is carried out in order to determine the stability and damping properties of the method besides a fairly reduced cpu effort numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method

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

a projection method for semidefinite linear systems and its applications we study the solution of consistent semidefinite and symmetric linear systems by iterative techniques given a finite sequence of subspaces a block iterative projection type algorithm is considered for two specific choices of iteration parameters we show convergence we apply our results to over and under determined linear equations these methods are based on decomposing the system matrix into blocks of rows or blocks of columns thereby several algorithms many used in image reconstruction are presented in a unified way

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

stage parallel fully implicit rungekutta solvers for discontinuous galerkin fluid simulations in this paper we develop new techniques for solving the large coupled linear systems that arise from fully implicit rungekutta methods this method makes use of the iterative preconditioned gmres algorithm for solving the linear systems which has seen success for fluid flow problems and discontinuous galerkin discretizations by transforming the resulting linear system of equations one can obtain a method which is much less computationally expensive than the untransformed formulation and which compares competitively with other time integration schemes such as diagonally implicit rungekutta dirk methods we develop and test several ilu based preconditioners effective for these large systems we additionally employ a parallel in time strategy to compute the rungekutta stages simultaneously numerical experiments are performed on the navierstokes equations using euler vortex and 2d and 3d naca airfoil test cases in serial and in parallel settings the fully implicit radau iia rungekutta methods compare favorably with equal order dirk methods in terms of accuracy number of gmres iterations number of matrixvector multiplications and wall clock time for a wide range of time steps

accelerated image reconstruction using ordered subsets of projection data the authors define ordered subset processing for standard algorithms such as expectation maximization em for image restoration from projections ordered subsets methods group projection data into an ordered sequence of subsets or blocks an iteration of ordered subsets em is defined as a single pass through all the subsets in each subset using the current estimate to initialize application of em with that data subset this approach is similar in concept to block kaczmarz methods introduced by eggermont et al 1981 for iterative reconstruction simultaneous iterative reconstruction sirt and multiplicative algebraic reconstruction mart techniques are well known special cases ordered subsets em os em provides a restoration imposing a natural positivity condition and with close links to the em algorithm os em is applicable in both single photon spect and positron emission tomography pet in simulation studies in spect the os em algorithm provides an order of magnitude acceleration over em with restoration quality maintained

noise characterization of block iterative reconstruction algorithms i theory researchers have shown increasing interest in block iterative image reconstruction algorithms due to the computational and modeling advantages they provide although their convergence properties have been well documented little is known about how they behave in the presence of noise in this work the authors fully characterize the ensemble statistical properties of the rescaled block iterative expectation maximization rbi em reconstruction algorithm and the rescaled block iterative simultaneous multiplicative algebraic reconstruction technique rbi smart also included in the analysis are the special cases of rbi em maximum likelihood em ml em and ordered subset em os em and the special case of rbi smart smart a theoretical formulation strategy similar to that previously outlined for ml em is followed for the rbi methods the theoretical formulations in this paper rely on one approximation namely that the noise in the reconstructed image is small compared to the mean image in a second paper the approximation will be justified through monte carlo simulations covering a range of noise levels iteration points and subset orderings the ensemble statistical parameters could then be used to evaluate objective measures of image quality

a high order solver for unsteady incompressible navier stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping we report development of a high order compact flux reconstruction method for solving unsteady incompressible flow on unstructured grids with implicit dual time stepping the method falls under the class of methods now referred to as flux reconstruction correction procedure via reconstruction the governing equations employ chorin s classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems an implicit non linear lower upper symmetric gauss seidel scheme with backward euler discretization is used to efficiently march the solution in pseudo time while a second order backward euler discretization is used to march in physical time we verify and validate implementation of the high order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems the current implicit time stepping scheme is proven effective in satisfying the divergence free constraint on the velocity field in the artificial compressibility formulation within the context of the high order flux reconstruction method this compact high order method is very suitable for parallel computing and can easily be extended to moving and deforming grids

robust implicit direct discontinuous galerkin method for simulating the compressible turbulent flows this paper proposes a robust and accurate implicit direct discontinuous galerkin ddg method for simulating laminar and turbulent flows the reynolds averaged navier stokes rans equations couple

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

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

a high order immersed boundary discontinuous galerkin method for poisson s equation with discontinuous coefficients and singular sources author s brandstetter g govindjee s abstract 2014 john wiley a sons ltd we adopt a numerical method to solve poisson s equation on a fixed grid with embedded boundary conditions where we put a special focus on the accurate representation of the normal gradient on the boundary the lack of accuracy in the gradient evaluation on the boundary is a common issue with low order embedded boundary methods whereas a direct evaluation of the gradient is preferable one typically uses post processing techniques to improve the quality of the gradient here we adopt a new method based on the discontinuous galerkin dg finite element method inspired by the recent work of a j lew and g c buscaglia a discontinuous galerkin based immersed boundary method international journal for numerical methods in engineering 76 427 454 2008 the method has been enhanced in two aspects firstly we approximate the boundary shape locally by higher order geometric primitives secondly we employ higher order shape functions within intersected elements these are derived for the various geometric features of the boundary based on analytical solutions of the underlying partial differential equation the development includes three basic geometric features in two dimensions for the solution of poisson s equation a straight boundary a circular boundary and a boundary with a discontinuity we demonstrate the performance of the method via analytical benchmark examples with a smooth circular boundary as well as in the presence of a singularity due to a re entrant corner results are compared to a low order extended finite element method as well as the dg method of 1 we report improved accuracy of the gradient on the boundary by one order of magnitude as well as improved convergence rates in the presence of a singular source in principle the method can be extended to three dimensions more complicated boundary shapes and other partial differential equations

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

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

an implicit high order hybridizable discontinuous galerkin method for nonlinear convection diffusion equations in this paper we present hybridizable discontinuous galerkin methods for the numerical solution of steady and time dependent nonlinear convection diffusion equations the methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary applying the newton raphson procedure and the hybridization technique we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every newton iteration the high number of globally coupled degrees of freedom in the discontinuous galerkin approximation is therefore significantly reduced we then extend the method to time dependent problems by approximating the time derivative by means of backward difference formulae when the time marching method is p 1 th order accurate and when polynomials of degree p 0 are used to represent the scalar variable each component of the flux and the approximate trace we observe that the approximations for the scalar variable and the flux converge with the optimal order of p 1 in the l 2 norm finally we apply element by element postprocessing schemes to obtain new approximations of the flux and the scalar variable the new approximate flux which has a continuous interelement normal component is shown to converge with order p 1 in the l 2 norm the new approximate scalar variable is shown to converge with order p 2 in the l 2 norm the postprocessing is performed at the element level and is thus much less expensive than the solution procedure for the time dependent case the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required extensive numerical results are provided to demonstrate the performance of the present method

a direct discontinuous galerkin method for the compressible navierstokes equations on arbitrary grids a direct discontinuous galerkin ddg method is developed for solving the compressible navierstokes equations on arbitrary grids in the framework of dg methods the ddg method originally introduced for scalar diffusion problems on structured grids is extended to discretize viscous and heat fluxes in the navierstokes equations two approaches of implementing the ddg method to compute numerical diffusive fluxes for the navierstokes equations are presented one is based on the conservative variables and the other is based on the primitive variables the importance of the characteristic cell size used in the ddg formulation on unstructured grids is examined the numerical fluxes on the boundary by the ddg method are discussed a number of test cases are presented to assess the performance of the ddg method for solving the compressible navierstokes equations based on our numerical results we observe that ddg method can achieve the designed order of accuracy and is able to deliver the same accuracy as the widely used br2 method at a significantly reduced cost clearly demonstrating that the ddg method provides an attractive alternative for solving the compressible navierstokes equations on arbitrary grids owning to its simplicity in implementation and its efficiency in computational cost

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

high order implicit hybridizable discontinuous galerkin methods for acoustics and elastodynamics we present a class of hybridizable discontinuous galerkin hdg methods for the numerical simulation of wave phenomena in acoustics and elastodynamics the methods are fully implicit and high order accurate in both space and time yet computationally attractive owing to their following distinctive features first they reduce the globally coupled unknowns to the approximate trace of the velocity which is defined on the element faces and single valued thereby leading to a significant saving in the computational cost in addition all the approximate variables including the approximate velocity and gradient converge with the optimal order of k 1 in the l 2 norm when polynomials of degree k 0 are used to represent the numerical solution and when the time stepping method is accurate with order k 1 when the time stepping method is of order k 2 superconvergence properties allows us by means of local postprocessing to obtain better yet inexpensive approximations of the displacement and velocity at any time levels for which an enhanced accuracy is required in particular the new approximations converge with order k 2 in the l 2 norm when k 1 for both acoustics and elastodynamics extensive numerical results are provided to illustrate these distinctive features

hybridizable discontinuous galerkin methods for the time harmonic maxwell s equations we present two hybridizable discontinuous galerkin hdg methods for the numerical solution of the time harmonic maxwell s equations the first hdg method explicitly enforces the divergence free condition and thus necessitates the introduction of a lagrange multiplier it produces a linear system for the degrees of freedom of the approximate traces of both the tangential component of the vector field and the lagrange multiplier the second hdg method does not explicitly enforce the divergence free condition and thus results in a linear system for the degrees of freedom of the approximate trace of the tangential component of the vector field only for both hdg methods the approximate vector field converges with the optimal order of k 1 in the l 2 norm when polynomials of degree k are used to represent all the approximate variables we propose elementwise postprocessing to obtain a new h c u r l conforming approximate vector field which converges with order k 1 in the h c u r l norm we present extensive numerical examples to demonstrate and compare the performance of the hdg methods

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

tight relaxations for polynomial optimization and lagrange multiplier expressions this paper proposes tight semidefinite relaxations for polynomial optimization the optimality conditions are investigated we show that generally lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points the polynomial expressions are determined by linear equations based on these expressions new lasserre type semidefinite relaxations are constructed for solving the polynomial optimization we show that the hierarchy of new relaxations has finite convergence or equivalently the new relaxations are tight for a finite relaxation order

a high order fully coupled upwind compact discontinuous galerkin method for modeling of viscous fingering in compressible porous media abstract we present a new approach for high fidelity compressible porous media simulations our method is based on a fully coupled upwind high order discontinuous galerkin formulation of the equations of miscible displacement transport a key feature of the proposed method is the high level of flexibility to compute complex subsurface geometries maintaining full order of accuracy the present formulation solves pressure and transport equations in a fully coupled fashion and as a result can capture the strong interaction between pressure and concentrations of injected components in highly compressible media the proposed method also shows very low sensitivity to mesh orientation in contrast with classical finite volume approximations used in porous media flow simulations the robustness and accuracy of the method are demonstrated in a number of challenging compressible and incompressible multiphase flow problems numerical simulations also reveal interesting phenomena associated with high pore compressibility such as the reduction of the growth rate of viscous fingers and the alteration of flow patterns near impermeable obstacles additional computations performed with an anisotropic permeability tensor reveal that somewhat against intuition viscous fingers have a preferential growth along the direction of low permeability

optimal runge kutta smoothers for the p multigrid discontinuous galerkin solution of the 1d euler equations this work presents a family of original runge kutta methods specifically designed to be effective relaxation schemes in the numerical solution of the steady state solution of purely advective problems with a high order accurate discontinuous galerkin space discretization and a p multigrid solution algorithm the design criterion for the construction of the runge kutta methods here developed is different form the one traditionally used to derive optimal runge kutta smoothers for the h multigrid algorithm which are designed in order to provide a uniform damping of the error modes in the high frequency range only the method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum the performance of the proposed schemes is assessed in the solution of the steady state quasi one dimensional euler equations for two test cases of increasing difficulty some preliminary results showing the performance for multidimensional applications are also presented

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

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

the convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations this paper reviews the convergence properties of linear stationary iterative processes for solving large unstructured systems of linear equations special attention is given to the case where the system to be solved is singular and possibly inconsistent the review starts by investigating the iteration bf x k 1 h bf x k bf f k 0 1 2 cdots when the system i h bf x bf f is inconsistent in this case the sequence left bf x k right always diverges nevertheless when h is convergent there exists a vector bf q in text null i h such that the sequence hat bf x k bf x k k bf q converges the limit of this sequence hat bf x solves a certain least squares problem and the sequence of residual vectors bf r k i h bf x k bf f converges to i h hat bf x bf f furthermore if h is similar to a real diagonal matrix then the corresponding norms of bf r k and hat bf x k hat bf x decr

accelerating the emml algorithm and related iterative algorithms by rescaled block iterative methods analysis of convergence of the algebraic reconstruction technique art shows it to be predisposed to converge to a solution faster than simultaneous methods such as those of the cimmino landweber 1992 type the expectation maximization maximum likelihood method for the poisson model emml and the simultaneous multiplicative art smart which use all the data at each step although the choice of ordering of the data and of relaxation parameters are important as herman and meyer 1993 have shown they are not the full story the analogous multiplicative art mart which applies only to systems y px in which y 0 p spl ges 0 and a nonnegative solution is sought is also sequential or row action rather than simultaneous but does not generally exhibit the same accelerated convergence relative to its simultaneous version smart by dividing each equation by the maximum of the corresponding row of p we find that this rescaled mart rmart does converge faster when solutions exist significantly so in cases in which the row maxima are substantially less than one such cases arise frequently in tomography and when the columns of p have been normalized to have sum one between simultaneous methods which use all the data at each step and sequential or row action methods which use only a single data value at each step there are the block iterative or ordered subset methods in which a single block or subset of the data is processed at each step the ordered subset em osem of hudson et al see ieee trans med imag vol 13 p 601 9 1994 is significantly faster than the emml but often fails to converge the rescaled block iterative emml rbi emml is an accelerated block iterative version of emml that converges in the consistent case to a solution for any choice of subsets it reduces to osem when the restrictive subset balanced condition holds rescaled block iterative versions of smart and mart also exhibit accelerated convergence

a fully coupled upwind discontinuous galerkin method for incompressible porous media flows high order computations of viscous fingering instabilities in complex geometry we present a new approach to the simulation of viscous fingering instabilities in incompressible miscible displacement flows in porous media in the past high resolution computational simulations of viscous fingering instabilities have always been performed using high order finite difference or fourier spectral methods which do not posses the flexibility to compute very complex subsurface geometries our approach instead by means of a fully coupled nonlinear implementation of the discontinuous galerkin method possesses a fundamental differentiating feature in that it maintains high order accuracy on fully unstructured meshes in addition the proposed method shows very low sensitivity to mesh orientation in contrast with classical finite volume approximation used in porous media flow simulations the robustness and accuracy of the method are demonstrated in a number of challenging computational problems

strong underrelaxation in kaczmarz s method for inconsistent systems we investigate the behavior of kaczmarz s method with relaxation for inconsistent systems we show that when the relaxation parameter goes to zero the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system this point minimizes the sum of the squares of the euclidean distances to the hyperplanes of the system if the starting point is chosen properly then the limits approach the minimum norm weighted least squares solution the proof is given for a block kaczmarz method

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 spalart allmaras turbulence model implementation in a discontinuous galerkin solver for incompressible flows in this paper the artificial compressibility flux discontinuous galerkin dg method for the solution of the incompressible navier stokes equations has been extended to deal with the reynolds averaged navier stokes rans equations coupled with the spalart allmaras sa turbulence model dg implementations of the rans and sa equations for compressible flows have already been reported in the literature including the description of limiting or stabilization techniques adopted in order to prevent the turbulent viscosity email protected from becoming negative in this paper we introduce an sa model implementation that deals with negative email protected values by modifying the source and diffusion terms in the sa model equation only when the working variable or one of the model closure functions become negative this results in an efficient high order implementation where either stabilization terms or even additional equations are avoided we remark that the proposed implementation is not dg specific and it is well suited for any numerical discretization of the rans sa governing equations the reliability robustness and accuracy of the proposed implementation have been assessed by computing several high reynolds number turbulent test cases the flow over a flat plate re 10 7 the flow past a backward facing step re 37400 and the flow around a naca 0012 airfoil at different angles of attack a 0 o 10 o 15 o and reynolds numbers re 2 88x10 6 6x10 6

iterative algorithms for large partitioned linear systems with applications to image reconstruction abstract we present a unifying framework for a wide class of iterative methods in numerical linear algebra in particular the class of algorithms contains kaczmarz s and richardson s methods for the regularized weighted least squares problem with weighted norm the convergence theory for this class of algorithms yields as corollaries the usual convergence conditions for kaczmarz s and richardson s methods the algorithms in the class may be characterized as being group iterative and incorporate relaxation matrices as opposed to a single relaxation parameter we show that some well known iterative methods of image reconstruction fall into the class of algorithms under consideration and are thus covered by the convergence theory we also describe a novel application to truly three dimensional image reconstruction

a hybridized discontinuous petrov galerkin scheme for scalar conservation laws summary we present a hybridized discontinuous petrov galerkin hdpg method for the numerical solution of steady and time dependent scalar conservation laws the method combines a hybridization technique with a local petrov galerkin approach in which the test functions are computed to maximize the inf sup condition since the petrov galerkin approach does not guarantee a conservative solution we propose to enforce this explicitly by introducing a constraint into the local petrov galerkin problem when the resulting nonlinear system is solved using the newton raphson procedure the solution inside each element can be locally condensed to yield a global linear system involving only the degrees of freedom of the numerical trace this results in a significant reduction in memory storage and computation time for the solution of the matrix system albeit at the cost of solving the local petrov galerkin problems however these local problems are independent of each other and thus perfectly scalable we present several numerical examples to assess the performance of the proposed method the results show that the hdpg method outperforms the hybridizable discontinuous galerkin method for problems involving discontinuities moreover for the test case proposed by peterson the hdpg method provides optimal convergence of order k 1 copyright 2012 john wiley sons ltd

choosing parameters in block iterative or ordered subset reconstruction algorithms viewed abstractly all the algorithms considered here are designed to provide a nonnegative solution x to the system of linear equations y px where y is a vector with positive entries and p a matrix whose entries are nonnegative and with no purely zero columns the expectation maximization maximum likelihood method as it occurs in emission tomography and the simultaneous multiplicative algebraic reconstruction technique are slow to converge on large data sets accelerating convergence through the use of block iterative or ordered subset versions of these algorithms is a topic of considerable interest these block iterative versions involve relaxation and normalization parameters the correct selection of which may not be obvious to all users the algorithms are not faster merely by virtue of being block iterative the correct choice of the parameters is crucial through a detailed discussion of the theoretical foundations of these methods we come to a better understanding of the precise roles these parameters play

polynomial optimization with real varieties we study the optimization problem of minimizing f x rm subject to h x 0 g x geq 0 with f a polynomial and h g two tuples of polynomials in x in mathbb r n lasserre s hierarchy is a sequence of sum of squares relaxations for finding the global minimum f min let k be the feasible set we prove the following results i if the real variety v mathbb r h is finite then lasserre s hierarchy has finite convergence no matter if the complex variety v mathbb c h is finite or not this solves an open question in the survey by m laurent ima vol math appl 149 springer new york 2009 pp 157 270 ii if k and v mathbb r h have the same vanishing ideal then the finite convergence of lasserre s hierarchy is independent of the choice of defining polynomials for the real variety v mathbb r h iii if only k is finite a refined version of lasserre s hierarchy using the preordering of g has finite 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

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

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

convergent block iterative algorithms for image reconstruction from inconsistent data it has been shown that convergence to a solution can be significantly accelerated for a number of iterative image reconstruction algorithms including simultaneous cimmino type algorithms the expectation maximization method for maximizing likelihood emml and the simultaneous multiplicative algebraic reconstruction technique smart through the use of rescaled block iterative bi methods these bi methods involve partitioning the data into disjoint subsets and using only one subset at each step of the iteration one drawback of these methods is their failure to converge to an approximate solution in the inconsistent case in which no image consistent with the data exists they are always observed to produce limit cycles lcs of distinct images through which the algorithm cycles no one of these images provides a suitable solution in general the question that arises then is whether or not these lc vectors retain sufficient information to construct from them a suitable approximate solution we show that they do to demonstrate that we employ a feedback technique in which the lc vectors are used to produce a new data vector and the algorithm restarted convergence of this nested iterative scheme to an approximate solution is then proven preliminary work also suggests that this feedback method may be incorporated in a practical reconstruction method

upwind methods for the baernunziato equations and higher order reconstruction using artificial viscosity this article is devoted to the numerical discretisation of the hyperbolic two phase flow model of baer and nunziato a special attention is paid on the discretisation of intercell flux functions in the framework of finite volume and discontinuous galerkin approaches where care has to be taken to efficiently approximate the non conservative products inherent to the model equations various upwind approximate riemann solvers have been tested on a bench of discontinuous test cases new discretisation schemes are proposed in a discontinuous galerkin framework following the criterion of abgrall and the path conservative formalism a stabilisation technique based on artificial viscosity is applied to the high order discontinuous galerkin method and compared against classical tvd muscl finite volume flux reconstruction

tensor eigenvalue complementarity problems this paper studies tensor eigenvalue complementarity problems basic properties of standard and complementarity tensor eigenvalues are discussed we formulate tensor eigenvalue complementarity problems as constrained polynomial optimization when one tensor is strictly copositive the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity when no tensor is strictly copositive we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process the complementarity eigenvalues can be computed sequentially the formulated polynomial optimization can be solved by lasserre s hierarchy of semidefinite relaxations we show that it has finite convergence for generic tensors numerical experiments are presented to show the efficiency of proposed methods

a unified analysis of balancing domain decomposition by constraints for discontinuous galerkin discretizations the bddc algorithm is extended to a large class of discontinuous galerkin dg discretizations of second order elliptic problems an estimate of c 1 log h h 2 is obtained for the condition number of the preconditioned system where c is a constant independent of h or h or large jumps in the coefficient of the problem numerical simulations are presented which confirm the theoretical results a key component for the development and analysis of the bddc algorithm is a novel perspective presenting the dg discretization as the sum of elementwise local bilinear forms the elementwise perspective allows for a simple unified analysis of a variety of dg methods and leads naturally to the appropriate choice for the subdomainwise local bilinear forms additionally this new perspective enables a connection to be drawn between the dg discretization and a related continuous finite element discretization to simplify the analysis of the bddc algorithm

an explicit hybridizable discontinuous galerkin method for the acoustic wave equation abstract we present an explicit hybridizable discontinuous galerkin hdg method for numerically solving the acoustic wave equation the method is fully explicit high order accurate in both space and time and coincides with the classic discontinuous galerkin dg method with upwinding fluxes for a particular choice of its stabilization function this means that it has the same computational complexity as other explicit dg methods however just as its implicit version it provides optimal convergence of order k 1 for all the approximate variables including the gradient of the solution and when the time stepping method is of order k 2 it displays a superconvergence property which allow us by means of local postprocessing to obtain new improved approximations of the scalar field variables at any time levels for which an enhanced accuracy is required in particular the new approximations converge with order k 2 in the l 2 norm for k 1 these properties do not hold for all numerical fluxes indeed our results show that when the hdg numerical flux is replaced by the lax friedrichs flux the above mentioned superconvergence properties are lost although some are recovered when the lax friedrichs flux is used only in the interior of the domain finally we extend the explicit hdg method to treat the wave equation with perfectly matched layers we provide numerical examples to demonstrate the performance of the proposed method

a time domain discontinuous galerkin method for mechanical resonator quality factor computations numerical simulations are becoming increasingly important in the design of micromechanical resonators in particular for the prediction of complex frequency response in high quality devices where damping is controlled by anchor losses frequency based approaches have been shown to predict these accurately however they require the solution of eigenvalue problems or the inversion of helmholtz type operators which are known to be very difficult for large scale iterative solvers we propose using a time domain approach instead where a broadband input signal is propagated through the system with a local explicit time stepper this is achieved using a new high order discontinuous galerkin dg discretization for the linear elasticity equations in particular a second order formulation with compact dg fluxes and a runge kutta time integrator where the block diagonal mass matrices allow for efficient stable and accurate time stepping our solver scales well on distributed parallel computers even in three spatial dimension and for large problem sizes the resulting output signal is analyzed using a well known filter diagonalization method which is capable of finding accurate frequencies and quality factors for as little as a hundred periods of data we validate the properties of our scheme on model problems and demonstrate the feasibility of our proposed analysis process on two high quality factor disk resonators using an axisymmetric formulation as well as full three dimensional simulations which is shown to scale well

the bassi rebay 1 scheme is a special case of the symmetric interior penalty formulation for discontinuous galerkin discretisations with gauss lobatto points abstract in the discontinuous galerkin dg community several formulations have been proposed to solve pdes involving second order spatial derivatives e g elliptic problems in this paper we show that when the discretisation is restricted to the usage of gauss lobatto points there are important similarities between two common choices the bassi rebay 1 br1 method and the symmetric interior penalty sip formulation this equivalence enables the extrapolation of properties from one scheme to the other a sharper estimation of the minimum penalty parameter for the sip stability compared to the more general estimate proposed by shahbazi 1 more efficient implementations of the br1 scheme and the compactness of the br1 method for straight quadrilateral and hexahedral meshes

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 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

an efficient time domain perfectly matched layers formulation for elastodynamics on spherical domains summary many practical applications require the analysis of elastic wave propagation in a homogeneous isotropic media in an unbounded domain one widely used approach for truncating the infinite domain is the so called method of perfectly matched layers pmls most existing pml formulations are developed for finite difference methods based on the first order velocity stress form of the elasticity equations and they are not straight forward to implement using standard finite element methods fems on unstructured meshes some of the problems with these formulations include the application of boundary conditions in half space problems and in the treatment of edges and or corners for time domain problems several pml formulations which do work with fems have been proposed although most of them still have some of these problems and or they require a large number of auxiliary nodal history memory variables in this work we develop a new pml formulation for time domain elastodynamics on a spherical domain which reduces to a two dimensional formulation under the assumption of axisymmetry our formulation is well suited for implementation using fems where it requires lower memory than existing formulations and it allows for natural application of boundary conditions we solve example problems on two dimensional and three dimensional domains using a high order discontinuous galerkin dg discretization on unstructured meshes and explicit time stepping we also study an approach for stabilization of the discrete equations and we show several practical applications for quality factor predictions of micromechanical resonators along with verifying the accuracy and versatility of our formulation copyright 2014 john wiley sons ltd

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

a reconstructed discontinuous galerkin method for the compressible navier stokes equations on arbitrary grids a reconstruction based discontinuous galerkin rdg method is presented for the solution of the compressible navier stokes equations on arbitrary grids the rdg method originally developed for the compressible euler equations is extended to discretize viscous and heat fluxes in the navier stokes equations using a so called inter cell reconstruction where a smooth solution is locally reconstructed using a least squares method from the underlying discontinuous dg solution similar to the recovery based dg rdg methods this reconstructed dg method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional dg methods unlike rdg methods this rdg method does not need to judiciously choose a proper form of a recovered polynomial thus is simple flexible and robust and can be used on arbitrary grids the developed rdg method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy efficiency robustness and versatility the numerical results indicate that this rdg method is able to deliver the same accuracy as the well known bassi rebay ii scheme at a half of its computing costs for the discretization of the viscous fluxes in the navier stokes equations clearly demonstrating its superior performance over the existing dg methods for solving the compressible navier stokes equations

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

reconstructed discontinuous galerkin methods for linear advection diffusion equations based on first order hyperbolic system abstract newly developed reconstructed discontinuous galerkin rdg methods are presented for solving linear advection diffusion equations on hybrid unstructured grids based on a first order hyperbolic system fohs formulation benefiting from both fohs and rdg methods the developed hyperbolic rdg methods are reliable accurate efficient and robust achieving higher orders of accuracy than conventional dg methods for the same number of degrees of freedom superior accuracy is achieved by reconstruction of higher order terms in the solution polynomial via gradient variables introduced to form a hyperbolic diffusion system and least squares variational reconstruction unsteady capability is demonstrated by an l stable implicit time integration scheme a number of advection diffusion test cases with a wide range of reynolds numbers including boundary layer type problems and unsteady cases are presented to assess accuracy and performance of the newly developed hyperbolic rdg methods numerical experiments demonstrate that the hyperbolic rdg methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular irregular and heterogeneous grids indicating that the developed hyperbolic rdg methods provide an attractive and probably an even superior alternative for solving the linear advection diffusion equations

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

efficiency of high order elements for continuous and discontinuous galerkin methods summary to evaluate the computational performance of high order elements a comparison based on operation count is proposed instead of runtime comparisons more specifically linear versus high order approximations are analyzed for implicit solver under a standard set of hypotheses for the mesh and the solution continuous and discontinuous galerkin methods are considered in two dimensional and three dimensional domains for simplices and parallelotopes moreover both element wise and global operations arising from different galerkin approaches are studied the operation count estimates show that for implicit solvers high order methods are more efficient than linear ones copyright 2013 john wiley sons ltd

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

stability of energy stable flux reconstruction for the diffusion problem using the interior penalty and bassi and rebay ii numerical fluxes for linear triangular elements abstract the flux reconstruction fr method has gained popularity within the research community the approach has been demonstrated to recover high order methods such as the discontinuous galerkin dg method stability analyses have been conducted for a linear advection problem leading to the energy stable flux reconstruction esfr methods also named vincent castonguay jameson huynh vcjh methods esfr schemes can be viewed as dg schemes with modally filtered correction fields using this class of methods the linear advection diffusion problem has been shown to be stable using the local discontinuous galerkin scheme ldg to compute the viscous numerical flux this stability proof has been extended for linear triangular and tetrahedra elements although the ldg scheme is commonly used it requires on particular meshes a wide stencil which raises the computational cost as a consequence many prefer the compact interior penalty ip or the bassi and rebay ii br2 numerical fluxes this article for the first time derives for both schemes a condition on the penalty term to ensure the stability for the linear diffusion equation the derivations are based on gauss legendre flux points and on periodic boundary conditions moreover the article proposes that for both the ip and br2 numerical fluxes the stability of the esfr scheme is independent of the auxiliary correction field a von neumann analysis is conducted to study the maximal time step of various esfr methods

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