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

an extension of the discontinuous galerkin method for the singular poisson equation we present a numerical method for solving a singular poisson equation which solution contains jumps and kinks due to a singular right hand side equations of this type may arise e g within the pressure computation of incompressible multiphase flows the method is an extension to the well known discontinuous galerkin dg method being able to represent the jumps and kinks with subcell accuracy in the proposed method an ansatz function which already fulfills the jump condition is subtracted from the original problem thereby reducing it to a standard poisson equation without a jump invoking a technique that we refer to as patching the construction of the ansatz function can be limited to a very narrow domain around the jump position thus making the construction numerically cheap and easy under optimal conditions the method shows a convergence order of p 1 for dg polynomial degree p still in the worst case a convergence order of approximately 2 4 is preserved for dg polynomial degree of 2

random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

extended discontinuous galerkin methods for two phase flows the spatial discretization summary this work discusses a discontinuous galerkin dg discretization for two phase flows the fluid interface is represented by a level set and the dg approximation space is adapted such that jumps and kinks in pressure and velocity fields can be approximated sharply this adaption of the original dg space which can be performed on the fly for arbitrary interface shapes is referred to as extended discontinuous galerkin by combining this ansatz with a special quadrature technique one can regain spectral convergence properties for low regularity solutions which is demonstrated by numerical examples this work focuses on the aspects of spatial discretization and special emphasis is devoted on how to overcome problems related to quadrature small cut cells and condition number of linear systems temporal discretization will be discussed in future works copyright 2016 john wiley sons ltd

from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex the electrostatic interpretation of the jacobi gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex moreover several new estimates based on extensive numerical studies for approximation along the line using jacobi gauss lobatto quadrature points as the nodal sets are presented the electrostatic analogy is extended to the two dimensional case with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented the matrices are evaluated by computing the lebesgue constants and they share the property that the nodes along the edges of the simplex are the gauss lobatto quadrature points of the chebyshev and legendre polynomials respectively this makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h p finite element methods

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

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 simple based discontinuous galerkin solver for steady incompressible flows in this paper we present how the well known simple algorithm can be extended to solve the steady incompressible navier stokes equations discretized by the discontinuous galerkin method the convective part is discretized by the local lax friedrichs fluxes and the viscous part by the symmetric interior penalty method within the simple algorithm the equations are solved in an iterative process the discretized equations are linearized and an equation for the pressure is derived on the discrete level the equations obtained for each velocity component and the pressure are decoupled and therefore can be solved sequentially leading to an efficient solution procedure the extension of the proposed scheme to the unsteady case is straightforward where fully implicit time schemes can be used various test cases are carried out the poiseuille flow the channel flow with constant transpiration the kovasznay flow the flow into a corner and the backward facing step flow using a mixed order formulation i e order k for the velocity and order k 1 for the pressure the scheme is numerically stable for all test cases convergence rates of k 1 and k in the l 2 norm are observed for velocity and pressure respectively a study of the convergence behavior of the simple algorithm shows that no under relaxation for the pressure is needed which is in strong contrast to the application of the simple algorithm in the context of the finite volume method or the continuous finite element method we conclude that the proposed scheme is efficient to solve the steady incompressible navier stokes equations in the context of the discontinuous galerkin method comprising hp accuracy

intrinsic volumes and f vectors of random polytopes on appelle polytope aleatoire k n l enveloppe convexe de n points choisis de facon aleatoire dans un corps convexe k on s interesse a l approximation stochastique de k par k n plus particulierement on obtient des resultats d approximation pour les esperances de v s k v s k n ou v s est le s ieme volume intrinseque du nombre de faces s dimensionnelles de k n et de la distance de hausdorff entre k et k n

central limit theorems for random polytopes let k be a smooth convex set the convex hull of independent random points in k is a random polytope central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity one essential step is to determine the precise asymptotic order of the occurring variances

the combinatorial structure of random polytopes abstract choose n random points in r d let p n be their convex hull and denote by f i p n the number of i dimensional faces of p n a general method for computing the expectation of f i p n i 0 d 1 is presented this generalizes classical results of efron in the case i 0 and renyi and sulanke in the case i d 1 to arbitrary i for random points chosen in a smooth convex body a limit law for f i p n is proved as n for random points chosen in a polytope the expectation of f i p n is determined as n this implies an extremal property for random points chosen in a simplex

skew symmetric nitsche s formulation in isogeometric analysis dirichlet and symmetry conditions patch coupling and frictionless contact abstract a simple skew symmetric nitsche s formulation is introduced into the framework of isogeometric analysis iga to deal with various problems in small strain elasticity essential boundary conditions symmetry conditions for kirchhoff plates patch coupling in statics and in modal analysis as well as signorini contact conditions for linear boundary or interface conditions the skew symmetric formulation is parameter free for contact conditions it remains stable and accurate for a wide range of the stabilization parameter several numerical tests are performed to illustrate its accuracy stability and convergence performance we investigate particularly the effects introduced by nitsche s coupling including the convergence performance and condition numbers in statics as well as the extra outlier frequencies and corresponding eigenmodes in structural dynamics we present the hertz test the block test and a 3d self contact example showing that the skew symmetric nitsche s formulation is a suitable approach to simulate contact problems in iga

explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements we derive new trace inequalities for nurbs mapped domains in addition to sobolev type inequalities we derive discrete trace inequalities for use in nurbs based isogeometric analysis all dependencies on shape size polynomial degree and the nurbs weighting function are precisely specified in our analysis and explicit values are provided for all bounding constants appearing in our estimates as hexahedral finite elements are special cases of nurbs our results specialize to parametric hexahedral finite elements and our analysis also generalizes to t spline based isogeometric analysis we compare the bounding constants appearing in our explicit trace inequalities with numerically computed optimal bounding constants and we discuss application of our results to a laplace problem we finish this paper with a brief exploration of so called patch wise trace inequalities for isogeometric analysis

fully computable error bounds for discontinuous galerkin finite element approximations on meshes with an arbitrary number of levels of hanging nodes we obtain fully computable a posteriori error bounds on the broken energy seminorm and discontinuous galerkin norm dg norm of the error in first order symmetric interior penalty galerkin sipg nonsymmetric interior penalty galerkin nipg and incomplete interior penalty galerkin iipg finite element approximations of a linear second order elliptic problem on meshes containing an arbitrary number of levels of hanging nodes and comprised of triangular elements the estimators are completely free of unknown constants and provide guaranteed numerical bounds on the broken energy seminorm and dg norm of the error these estimators are also shown to provide a lower bound for the broken energy seminorm and dg norm of the error up to a constant and higher order data oscillation terms we also obtain an explicit computable bound for the value of the interior penalty parameter needed to ensure the existence of the discontinuous galerkin finite element approximation for all versions of the method

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

h p finite element trace inequalities for the pyramid polynomial trace inequalities typically involve an unknown constant depending on the order n of the polynomial the dependence of these constants on n was made more explicit in warburton and hesthaven 2003 for the general d simplex and they were determined for quadrilaterals hexes and wedges in hillewaert 2013 in this note we derive explicit expressions for the constant in the trace inequality for two sets of basis functions on the pyramid

smoothed aggregation multigrid for the discontinuous galerkin method the aim of this paper is to investigate theoretically as well as experimentally an algebraic multilevel algorithm for the solution of the linear systems that arise from the discontinuous galerkin method the smoothed aggregation multigrid introduced by vanek for the conforming finite element method is applied to low order discretizations of convection diffusion equations for the elliptic model problem the algorithm is shown to be quasi optimal adjustments for the case of nonvanishing advection such as directionally implicit smoothing and a suitable splitting of the operator are discussed several numerical experiments are presented for two dimensional problems including a newton type linearization of the compressible navier stokes equations

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

kakutani property of the polytopes implies kakutani property of the whole space abstract this paper deals with the following property of a space x for every upper semicontinuous function from x to nonempty closed convex subsets of x there exists x 0 which is in x 0 we derive this property of x from the same property assumed to hold for the polytopes in x convexity in our setup is given in an abstract axiomatic way the results are also reformulated in the order theoretical language of continuous lattices our results directly generalize a an extension of the kakutani fixed point theorem to compact convex sets in locally convex spaces due to ky fan and i l glicksberg and b the fixed point theorem of wallace concerning set valued functions on trees

on the stability of the symmetric interior penalty method for the spalart allmaras turbulence model the stability and convergence of the interior penalty ip discontinuous galerkin method are closely related to the penalty coefficient s f using sharp trace inequalities adapted to the functional space shahbazi 2005 7 has derived optimal values of s f for constant diffusivity problems on triangular meshes we propose a generalisation of his analysis to account for mesh anisotropy and different element types on the one hand and strong variations of the diffusivity on the other which characterise the spalart allmaras rans turbulence model the adequacy of this new definition is illustrated by applications to benchmark 2d computations finally a comparison with two state of the art finite volume solvers is presented for a 3d high lift cascade flow

on the stability of projection methods for the incompressible navier stokes equations based on high order discontinuous galerkin discretizations abstract the present paper deals with the numerical solution of the incompressible navier stokes equations using high order discontinuous galerkin dg methods for discretization in space for dg methods applied to the dual splitting projection method instabilities have recently been reported that occur for small time step sizes since the critical time step size depends on the viscosity and the spatial resolution these instabilities limit the robustness of the navier stokes solver in case of complex engineering applications characterized by coarse spatial resolutions and small viscosities by means of numerical investigation we give evidence that these instabilities are related to the discontinuous galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high order accuracy with respect to the temporal discretization this new formulation is stable in the limit of small time steps for both equal order and mixed order polynomial approximations although the dual splitting scheme itself includes inf sup stabilizing contributions we demonstrate that spurious pressure oscillations appear for equal order polynomials and small time steps highlighting the necessity to consider inf sup stability explicitly

discontinuous galerkin isogeometric analysis for the biharmonic equation abstract we present and analyze an interior penalty discontinuous galerkin isogeometric analysis dg iga method for the biharmonic equation in computational domain in r d with d 2 3 the computational domain consists of several non overlapping sub domains or patches we construct b spline approximation spaces which are discontinuous across patch interfaces we present a priori error estimate in a discrete norm and numerical experiments to confirm the theory

variational inequality approach to enforcing the non negative constraint for advection diffusion equations abstract predictive simulations are crucial for the success of many subsurface applications and it is highly desirable to obtain accurate non negative solutions for transport equations in these numerical simulations to this end optimization based methodologies based on quadratic programming qp have been shown to be a viable approach to ensuring discrete maximum principles and the non negative constraint for anisotropic diffusion equations in this paper we propose a computational framework based on the variational inequality vi which can also be used to enforce important mathematical properties e g maximum principles and physical constraints e g the non negative constraint we demonstrate that this framework is not only applicable to diffusion equations but also to non symmetric advection diffusion equations an attractive feature of the proposed framework is that it works with any weak formulation for the advection diffusion equations including single field formulations which are computationally attractive a particular emphasis is placed on the parallel and algorithmic performance of the vi approach across large scale and heterogeneous problems it is also shown that qp and vi are equivalent under certain conditions state of the art qp and vi solvers available from the petsc library are used on a variety of steady state 2d and 3d benchmarks and a comparative study on the scalability between the qp and vi solvers is presented we then extend the proposed framework to transient problems by simulating the miscible displacement of fluids in a heterogeneous porous medium and illustrate the importance of enforcing maximum principles for these types of coupled problems our numerical experiments indicate that vis are indeed a viable approach for enforcing the maximum principles and the non negative constraint in a large scale computing environment also provided are firedrake project files as well as a discussion on the computer implementation to help facilitate readers in understanding the proposed framework

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

optimized multiplicative additive and restricted additive schwarz preconditioning we demonstrate that a small modification of the multiplicative additive and restricted additive schwarz preconditioner at the algebraic level motivated by optimized schwarz methods defined at the continuous level leads to a significant reduction in the iteration count of the iterative solver numerical experiments using finite difference and spectral element discretizations of the positive definite helmholtz problem and an idealized climate simulation illustrate the effectiveness of the new approach

a discontinuous galerkin method for the mono energetic fokker planck equation based on a spherical interior penalty formulation we present a new discretization of the mono energetic fokker planck equation we build on previous work kophazi and lathouwers 2015 where we devised an angular discretization for the boltzmann equation allowing for both heterogeneous and anisotropic angular refinement the angular discretization is based on a discontinuous finite element method on the unit sphere here we extend the methodology to include the effect of the fokker planck scatter operator describing small angle particle scatter we describe the construction of an interior penalty method on the sphere surface results are provided for a variety of test cases ranging from purely angular to fully three dimensional the results show that the scheme can resolve highly forward peaked flux distributions with forward peaked scatter

a selection theorem for topological convex structures a continuous selection theorem has been obtained for multivalued functions the values of which are convex sets of certain synthetic convex structures applications are given related with superextensions semi lattices spaces of order arcs trees whitney levels in hyperspaces and geometric topology applications to traditional convexity in vector spaces involve beer s approximation theorem and a fixed point theorem of dugundji granas some other applications a o an invariant arc theorem appear elsewhere

a discontinuous continuous low order finite element shallow water model on the sphere we study the applicability of a new finite element in atmosphere and ocean modeling the finite element under investigation combines a second order continuous representation for the scalar field with a first order discontinuous representation for the velocity field and is therefore different from continuous and discontinuous galerkin finite element approaches the specific choice of low order approximation spaces is attractive because it satisfies the ladyzhenskaya babuska brezzi condition and is at the same time able to represent the crucially important geostrophic balance the finite element is used to solve the viscous and inviscid shallow water equations on a rotating sphere we introduce the spherical geometry via a stereographic projection the projection leads to a manageable number of additional terms the associated scaling factors can be exactly represented by second order polynomials we perform numerical experiments considering steady and unsteady zonal flow flow over topography and an unstable zonal jet stream for ocean applications the wind driven stommel gyre is simulated the experiments are performed on icosahedral geodesic grids and analyzed with respect to convergence rates conservation properties and energy and enstrophy spectra the results match quite well with results published in the literature and encourage further investigation of this type of element for three dimensional atmosphere ocean modeling

on the constants in hp finite element trace inverse inequalities we derive inverse trace inequalities for hp finite elements utilizing orthogonal polynomials we show how to derive explicit expressions for the constants when considering triangular and tetrahedral elements we also discuss how to generalize this technique to the general d simplex

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

a posteriori error estimator based on gradient recovery by averaging for discontinuous galerkin methods we consider some anisotropic and piecewise constant diffusion problems in domains of r 2 approximated by a discontinuous galerkin method with polynomials of any fixed degree we propose an a posteriori error estimator based on gradient recovery by averaging it is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability the lower bound is also established moreover these additional terms are negligible when the recovered gradient is superconvergent the reliability and efficiency of the proposed estimator is confirmed by some numerical tests

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

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

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

on the variance of random polytopes abstract a random polytope is the convex hull of uniformly distributed random points in a convex body k a general lower bound on the variance of the volume and f vector of random polytopes is proved also an upper bound in the case when k is a polytope is given for polytopes as for smooth convex bodies the upper and lower bounds are of the same order of magnitude the results imply a law of large numbers for the volume and f vector of random polytopes when k is a polytope

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

penalty factor threshold and time step bound estimations for discontinuous galerkin time domain method based on helmholtz equation in this article penalty factor threshold and time step bound in discontinuous galerkin time method based on vector wave equation dgtd we method are well estimated based on the semidiscrete form of the dgtd we method properties of the system matrices are studied and the stability condition related to the penalty factor is derived by decomposing the global system matrices of the dgtd we method into the local ones and developing an efficient iteration procedure the lower bound threshold of the penalty factor is well estimated to guarantee the positive semidefinite property of the global system matrices with the calculated penalty factor the maximum time step is analytically determined by approximating spectral radius of the local system matrix both the penalty factor bound and the maximal time step are computed element wise instead of a global system matrix operation and thus the proposed method can be efficiently applied into the large scale meshes with different types of the basis functions and boundary conditions numerical examples are presented to demonstrate the validity and good performance of the proposed methods

multi patch discontinuous galerkin isogeometric analysis for wave propagation explicit time stepping and efficient mass matrix inversion abstract we present a class of spline finite element methods for time domain wave propagation which are particularly amenable to explicit time stepping the proposed methods utilize a discontinuous galerkin discretization to enforce continuity of the solution field across geometric patches in a multi patch setting which yields a mass matrix with convenient block diagonal structure over each patch we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight adjusted approximation of the mass matrix inverse this approximation restores a tensor product structure while retaining provable high order accuracy and semi discrete energy stability we also estimate the maximum stable timestep for spline based finite elements and show that the use of spline spaces results in less stringent cfl restrictions than equivalent c 0 or discontinuous finite element spaces finally we explore the use of optimal knot vectors based on l 2 n widths we show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep and present a simple heuristic method for approximating optimal knot positions numerical experiments confirm the accuracy and stability of the proposed methods

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

random polytopes in a convex polytope independence of shape and concentration of vertices write d for the set of all convex bodies convex compact sets with nonempty interior in d define o g l d as the set of those k e 5 b d with vol k 1 fix k e g i d and choose points x l x e k randomly independently and according to the uniform distribution on k then k c o n v x l xn is a random polytope in k write e k n for the expectation of the random variable v o l k k n e k n shows how well k approximates k in volume on the average groemer grl proved that among all convex bodies k e o g l d the ellipsoids are approximated worst i e

a high order semi explicit discontinuous galerkin solver for 3d incompressible flow with application to dns and les of turbulent channel flow abstract we present an efficient discontinuous galerkin scheme for simulation of the incompressible navier stokes equations including laminar and turbulent flow we consider a semi explicit high order velocity correction method for time integration as well as nodal equal order discretizations for velocity and pressure the non linear convective term is treated explicitly while a linear system is solved for the pressure poisson equation and the viscous term the key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under resolved high reynolds number flows as well as small time steps this penalty method is similar to the grad div stabilization widely used in continuous finite elements we further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations the solver is specifically designed for large scale computations through matrix free linear solvers including efficient preconditioning strategies and tensor product elements which have allowed us to scale this code up to 34 4 billion degrees of freedom and 147 456 cpu cores we validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large eddy simulation of turbulent channel flow at r e 180 as well as 590

mixed hp dgfem for incompressible flows we consider several mixed discontinuous galerkin approximations of the stokes problem and propose an abstract framework for their analysis using this framework we derive a priori error estimates for hp approximations on tensor product meshes we also prove a new stability estimate for the discrete divergence bilinear form

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

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