on the computation of steady state compressible flows using a discontinuous galerkin method computation of compressible steady state flows using a high order discontinuous galerkin finite element method is presented in this paper an accurate representation of the boundary normals based on the definition of the geometries is used for imposing solid wall boundary conditions for curved geometries particular attention is given to the impact and importance of slope limiters on the solution accuracy for flows with strong discontinuities a physics based shock detector is introduced to effectively make a distinction between a smooth extremum and a shock wave a recently developed fast low storage p multigrid method is used for solving the governing compressible euler equations to obtain steady state solutions the method is applied to compute a variety of compressible flow problems on unstructured grids numerical experiments for a wide range of flow conditions in both 2d and 3d configurations are presented to demonstrate the accuracy of the developed discontinuous galerkin method for computing compressible steady state flows

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

a p multigrid discontinuous galerkin method for the euler equations on unstructured grids a p multigrid p polynomial degree discontinuous galerkin method is presented for the solution of the compressible euler equations on unstructured grids the method operates on a sequence of solution approximations of different polynomial orders a distinct feature of this p multigrid method is to use different time integration schemes on different approximation levels resulting in an accurate fast and low memory method that can be used to accelerate the convergence of the euler equations to a steady state for discontinuous galerkin methods the developed method is used to compute the compressible flows for a variety of test problems on unstructured grids the numerical results obtained strongly indicate the order independent property of this p multigrid method an overall speed up factor more than one order of magnitude for both second and third order solutions of all test cases in comparison with the explicit method is demonstrated

fast p multigrid discontinuous galerkin method for compressible flows at all speeds ap multigrid wherep is the polynomial degree discontinuous galerkin method is presented for the solution of the compressible euler equations on unstructured grids the method operates on a sequence of solution approximations of different polynomial orders a distinct feature ofthisp multigrid method is the application of an explicit smoother on the higher level approximations p 0 and an implicit smoother on the lowest level approximation p 0 resulting in a fast and low storage method that can be efficiently used to accelerate the convergence to a steady state solution furthermore this p multigrid method can be naturally applied to compute the flows with discontinuities in which a monotonic limiting procedure is usually required for discontinuous galerkin methods an accurate representation of the boundary normals based on the definition of the geometries is used for imposing slip boundary conditions for curved geometries krivodonova l and berger m high order implementation of solid wall boundary conditions in curved geometries journal of computational physics vol 211 no 2 2006 pp 492 512 and luo h baum j d and lohner r on the computation of steady state compressible flows using a discontinuous galerkin method international journal for numerical methods in engineering vol 73 no 5 2008 pp 597 623 a variety of compressible flow problems for a wide range of flow conditions from low mach number to supersonic in both two dimensional and three dimensional configurations are computed to demonstrate the performance ofthisp multigrid method the numerical results obtained strongly indicate the order independent property of this p multigrid method and demonstrate that this method is orders of magnitude faster than its explicit counterpart the performance comparison with a finite volume method shows that using this p multigrid method the discontinuous galerkin method provides a viable attractive competitive and probably even superior alternative to the finite volume method for computing compressible flows at all speeds

the generation of triangular meshes for nurbs enhanced fem summary this paper presents the first method that enables the fully automatic generation of triangular meshes suitable for the so called non uniform rational b spline nurbs enhanced finite element method nefem the meshes generated with the proposed approach account for the computer aided design boundary representation of the domain given by nurbs curves the characteristic element size is completely independent of the geometric complexity and of the presence of very small geometric features the proposed strategy allows to circumvent the time consuming process of de featuring complex geometric models before a finite element mesh suitable for the analysis can be produced a generalisation of the original definition of a nefem element is also proposed enabling to treat more complicated elements with an edge defined by several nurbs curves or more than one edge defined by different nurbs three examples of increasing difficulty demonstrate the applicability of the proposed approach and illustrate the advantages compared with those of traditional finite element mesh generators finally a simulation of an electromagnetic scattering problem is considered to show the applicability of the generated meshes for finite element analysis 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

solving discontinuous galerkin formulations of poisson s equation using geometric and p multigrid methods for solving discontinuous galerkin formulations of the poisson equation by coupling p multigrid to geometric multigrid are investigated the simple approach of performing iterative relaxation on solution approximations of decreasing polynomial degree p down to p 0 and then applying geometric multigrid is ineffective the transition from p 1 top 0 causes the performance of the entire iteration to degrade because the long wavelength eigenfunctions of the p 1 discontinuous system are not represented well in thep 0 space a new approach is proposed that coarsens from thep 1 discontinuous space to thep 1 continuous space this approach eliminates the problems caused by the p 1 to p 0 transition furthermore the p 1 continuous space is a standard finite element space for which geometric multigrid is well defined in addition when the discontinuous galerkin equations are restricted to a continuous space one recovers the galerkin formulation of continuous finite elements thus applying geometric multigrid to this system is straightforward and effective numerical tests agree well with the analysis and confirm that the new approach gives rapid convergence rates that are grid independent and only weakly sensitive to the polynomial order

lower upper implicit schemes with multiple grids for the euler equations on montre qu une methode de surrelaxation successive pour la resolution du schema implicite non factorise est equivalente a une forme modifiee du schema implicite inferieur superieur

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

a hermite weno reconstruction based discontinuous galerkin method for the euler equations on tetrahedral grids a hermite weno reconstruction based discontinuous galerkin method rdg p 1p 2 designed not only to enhance the accuracy of discontinuous galerkin method but also to ensure linear stability of the rdg method is presented for solving the compressible euler equations on tetrahedral grids in this rdg p 1p 2 method a quadratic polynomial solution p 2 is first reconstructed using a least squares method from the underlying linear polynomial p 1 discontinuous galerkin solution by taking advantage of handily available and yet invaluable information namely the derivatives in the dg formulation the stencils used in the reconstruction involve only von neumann neighborhood adjacent face neighboring cells and thus are compact and consistent with the underlying dg method the final quadratic polynomial solution is then obtained using a weno reconstruction which is necessary to ensure linear stability of the rdg method the developed rdg method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy efficiency robustness and versatility the numerical experiments demonstrate that the developed rdg p 1p 2 method is able to maintain the linear stability achieve the designed third order of accuracy one order accuracy higher than the underlying dg method without significant increase in computing costs and storage requirements

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

a reconstructed discontinuous galerkin method based on a hierarchical weno reconstruction for compressible flows on tetrahedral grids a reconstructed discontinuous galerkin rdg method based on a hierarchical weno reconstruction termed hweno p 1p 2 in this paper designed not only to enhance the accuracy of discontinuous galerkin methods but also to ensure the nonlinear stability of the rdg method is presented for solving the compressible euler equations on tetrahedral grids in this hweno p 1p 2 method a quadratic polynomial solution p 2 is first reconstructed using a hermite weno reconstruction from the underlying linear polynomial p 1 discontinuous galerkin solution to ensure the linear stability of the rdg method and to improve the efficiency of the underlying dg method by taking advantage of handily available and yet invaluable information namely the derivatives in the dg formulation the stencils used in the reconstruction involve only von neumann neighborhood adjacent face neighboring cells and thus are compact the first derivatives of the quadratic polynomial solution are then reconstructed using a weno reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities thus ensuring the nonlinear stability of the rdg method the developed hweno p 1p 2 method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy robustness and non oscillatory property the numerical experiments indicate that the hweno p 1p 2 method is able to capture shock waves within one cell without any spurious oscillations and achieve the designed third order of accuracy one order accuracy higher than the underlying dg method

a third order partial differential equation for isotropic boundary based triangular bezier surface generation abstract we approach surface design by solving a linear third order partial differential equation pde we present an explicit polynomial solution method for triangular bezier pde surface generation characterized by a boundary configuration the third order pde comes from a symmetric operator defined here to overcome the anisotropy drawback of any operator over triangular bezier surfaces

discontinuous galerkin finite element method for euler and navier stokes equations a finite element method for the euler and navier stokes equations has been developed the spatial discretization involves the discontinuous galerkin finite element method and lax friedrichs flux method the temporal discretizations used are the explicit runge kutta time integrations the scheme is formally second order accurate in space and time a dynamic mesh algorithm is included to simulate flows over moving bodies the inviscid flows passing through a channel with circular arc bump through the naca 0012 airfoil and the laminar flows passing over a flat plate with shock interaction are investigated to confirm the accuracy and convergence of the finite element method also the unsteady flow through a pitching naca 0012 airfoil is performed to prove the capability of the present method

a reconstructed discontinuous galerkin method for magnetohydrodynamics on arbitrary grids a reconstructed discontinuous galerkin rdg method designed not only to enhance the accuracy of dg methods but also to ensure the nonlinear stability of the rdg method is developed for solving the magnetohydrodynamics mhd equations on arbitrary grids in this rdg p1p2 method a quadratic polynomial solution p2 is first obtained using a hermite weighted essentially non oscillatory weno reconstruction from the underlying linear polynomial p1 discontinuous galerkin solution to ensure linear stability of the rdg method and to improves efficiency of the underlying dg method by taking advantage of handily available and yet invaluable information namely the first derivatives in the dg formulation the stencils used in reconstruction involve only von neumann neighborhood adjacent face neighboring cells and thus are compact the first derivatives of the quadratic polynomial solution are then reconstructed using a weno reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities thus ensuring the nonlinear stability of the rdg method the hlld riemann solver introduced in the literature for one dimensional mhd problems is adopted in normal direction to compute numerical fluxes the divergence free constraint is satisfied using the locally divergence free ldf approach the developed rdg method is used to compute a variety of 2d and 3d mhd problems on arbitrary grids to demonstrate its accuracy robustness and non oscillatory property our numerical experiments indicate that the rdg p1p2 method is able to capture shock waves sharply essentially without any spurious oscillations and achieve the designed third order of accuracy one order accuracy higher than the underlying dg method

p multigrid methods and their comparison to h multigrid methods within isogeometric analysis abstract over the years isogeometric analysis has shown to be a successful alternative to the finite element method fem however solving the resulting linear systems of equations efficiently remains a challenging task in this paper we consider a p multigrid method in which coarsening is applied in the spline degree p instead of the mesh width h and compare it to h multigrid methods since the use of classical smoothers e g gauss seidel results in a p multigrid h multigrid method with deteriorating performance for higher values of p the use of an ilut smoother is investigated as well numerical results and a spectral analysis indicate that the use of this smoother exhibits convergence rates essentially independent of h and p for both p multigrid and h multigrid methods in particular we compare both coarsening strategies e g coarsening in h or p adopting both smoothers for a variety of two and three dimensional benchmarks furthermore the ilut smoother is compared to a state of the art smoother hofreither and takacs 2017 using both coarsening strategies finally the proposed p multigrid method is used to solve linear systems resulting from thb spline discretizations

harnack inequalities and gaussian estimates for a class of hypoelliptic operators we prove a global harnack inequality for a class of degenerate evolution operators by using repeatedly an invariant local harnack inequality as a consequence we obtain an accurate gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators

coupling p multigrid to geometric multigrid for discontinuous galerkin formulations of the convection diffusion equation an improved p multigrid algorithm for discontinuous galerkin dg discretizations of convection diffusion problems is presented the general p multigrid algorithm for dg discretizations involves a restriction from the p 1 to p 0 discontinuous polynomial solution spaces this restriction is problematic and has limited the efficiency of the p multigrid method for purely diffusive problems helenbrook and atkins have demonstrated rapid convergence using a method that restricts from a discontinuous to continuous polynomial solution space at p 1 it is shown that this method is not directly applicable to the convection diffusion cd equation because it results in a central difference discretization for the convective term to remedy this ideas from the streamwise upwind petrov galerkin supg formulation are used to devise a transition from the discontinuous to continuous space at p 1 that yields an upwind discretization the results show that the new method converges rapidly for all peclet numbers

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

scale separation in fast hierarchical solvers for discontinuous galerkin methods we present a method for solution of linear systems resulting from discontinuous galerkin dg approximations the two level algorithm is based on a hierarchical scale separation scheme hss such that the linear system is solved globally only for the cell mean values which represent the coarse scales of the dg solution the system matrix of this coarse scale problem is exactly the same as in the cell centered finite volume method the higher order components of the solution fine scales are computed as corrections by solving small local problems this technique is particularly efficient for dg schemes that employ hierarchical bases and leads to an unconditionally stable method for stationary and time dependent hyperbolic and parabolic problems unlike p multigrid schemes only two levels are used for dg approximations of any order the proposed method is conceptually simple and easy to implement it compares favorably to p multigrid in our numerical experiments numerical tests confirm the accuracy and robustness of the proposed algorithm

acceleration for microflow simulations of high order moment models by using lower order model correction we study the acceleration of steady state computation for microflow which is modeled by the high order moment models derived recently from the steady state boltzmann equation with bgk type collision term by using the lower order model correction a novel nonlinear multi level moment solver is developed numerical examples verify that the resulting solver improves the convergence significantly thus is able to accelerate the steady state computation greatly the behavior of the solver is also numerically investigated it is shown that the convergence rate increases indicating the solver would be more efficient as the total levels increases three order reduction strategies of the solver are considered numerical results show that the most efficient order reduction strategy would be ml1 ml 2

pde triangular bezier surfaces harmonic biharmonic and isotropic surfaces we approach surface design by solving second order and fourth order partial differential equations pdes we present many methods for designing triangular bezier pde surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces moreover we introduce and study a second order and a fourth order symmetric operator to overcome the anisotropy drawback of the harmonic and biharmonic operators over triangular bezier surfaces