To CG or to HDG: A Comparative Study

Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called static condensation in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement our HDG approach within a spectral/hp element framework so that comparisons can be done between HDG and the traditional CG approach.     

Arbitrary High-Order Explicit Hybridizable Discontinuous Galerkin Methods for the Acoustic Wave Equation

We propose a new formulation of explicit time integration for the hybridizable discontinuous Galerkin (HDG) method in the context of the acoustic wave equation based on the arbitrary derivative approach. The method is of arbitrary high order in space and time without restrictions such as the Butcher barrier for Runge–Kutta methods. To maintain the superconvergence property characteristic for HDG spatial discretizations, a special reconstruction step is developed, which is complemented by an adjoint consistency analysis. For a given time step size, this new method is twice as fast compared to a low-storage Runge–Kutta scheme of order four with five stages at polynomial degrees between two and four. Several numerical examples are performed to demonstrate the convergence properties, reveal dispersion and dissipation errors, and show solution behavior in the presence of material discontinuities. Also, we present the combination of local time stepping with h-adaptivity on three-dimensional meshes with curved elements.     

A Posteriori Error Estimates for HDG Methods

We present the first a posteriori error analysis of the so-called hybridizable discontinuous Galerkin (HDG) methods for second-order elliptic problems. We show that the error in the flux can be controlled by only two terms. The first term captures the so-called data oscillation. The second solely depends on the difference between the trace of the scalar approximation and the corresponding numerical trace. Numerical experiments verifying the reliability and efficiency of the estimate in two-space dimensions are presented.     

Dispersion Analysis of HDG Methods

This work presents a dispersion analysis of the Hybrid Discontinuous Galerkin (HDG) method. Considering the Helmholtz system, we quantify the discrepancies between the exact and discrete wavenumbers. In particular, we obtain an analytic expansion for the wavenumber error for the lowest order Single Face HDG (SFH) method. The expansion shows that the SFH method exhibits convergence rates of the wavenumber errors comparable to that of the mixed hybrid Raviart–Thomas method. In addition, we observe the same behavior for the higher order cases in numerical experiments.     

Computer contouring of sea bed topography and finite element simulation of tidal flow in Johor Strait

Computer contouring of sea-bed topography is presented based on Kriging method for the Johor Strait. This automates the input of water depths into a finite element tidal model which is formulated on the shallow water theory. The Galerkin weighted residual technique and the two-step Lax-Wendroff scheme are applied to discretize the vertically integrated governing equations and to advance the numerical solutions in time respectively. The model is assessed against field data. There is a fair degree of agreement between the simulation results and the actual field measurements.     

Bifurcations and internal resonances in space-curved rods

The paper discusses a geometrically non-linear model of space-curved beams. The Timoshenko-type model is used, which includes the shear effects and rotatory inertia. The motion of the system is described by a non-linear matrix equation, which accounts for non-linearities up to the second order. To solve the non-linear problem, discretization methods based on the Galerkin method are used and the resulting non-linear eigenvalue problem is then solved by the continuation methods. The analysis of the discretized problem allows the study of the bifurcations of solutions. The post-bifurcation analysis is used to explain the phenomenon of internal resonance, which is of much importance in the prediction of the dynamic response of the analyzed systems. The approach is illustrated by numerical examples which consider the free vibration of beams and circular arches with different boundary conditions.     

Spatial approximation of the radiation transport equation using a subgrid-scale finite element method

In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition of the unknowns into resolvable and subgrid scales, with an approximation for the latter that yields a problem to be solved for the former. This approach allows us to design the algorithmic parameters on which the method depends, which we do here when the discrete ordinates method is used for the directional approximation. We concentrate on two stabilized methods, namely, the classical SUPG technique and the orthogonal subscale stabilization. A numerical analysis of the spatial approximation for both formulations is performed, which shows that they have a similar behavior: they are both stable and optimally convergent in the same mesh-dependent norm. A comparison with the behavior of the Galerkin method, for which a non-standard numerical analysis is done, is also presented.     

Local Discontinuous Galerkin Finite Element Method and Error Estimates for One Class of Sobolev Equation

In this paper we present a numerical scheme based on the local discontinuous Galerkin (LDG) finite element method for one class of Sobolev equations, for example, generalized equal width Burgers equation. The proposed scheme will be proved to have good numerical stability and high order accuracy for arbitrary nonlinear convection flux, when time variable is continuous. Also an optimal error estimate is obtained for the fully discrete scheme, when time is discreted by the second order explicit total variation diminishing (TVD) Runge-Kutta time-marching. Finally some numerical results are given to verify our analysis for the scheme.     

A priori and a posteriori error analyses of an HDG method for the Brinkman problem

In this paper we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for the linear Brinkman model of porous media flow in two and three dimensions and with non-homogeneous Dirichlet boundary conditions. We consider a fully-mixed formulation in which the main unknowns are given by the pseudostress, the velocity and the trace of the velocity, whereas the pressure is easily recovered through a simple postprocessing. We show that the corresponding continuous and discrete schemes are well-posed. In particular, we use the projection-based error analysis in order to derive a priori error estimates. Furthermore, we develop a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator and showing the expected behavior of the adaptive refinements are presented.     

Exploiting Batch Processing on Streaming Architectures to Solve 2D Elliptic Finite Element Problems: A Hybridized Discontinuous Galerkin (HDG) Case Study

Numerical methods for elliptic partial differential equations (PDEs) within both continuous and hybridized discontinuous Galerkin (HDG) frameworks share the same general structure: local (elemental) matrix generation followed by a global linear system assembly and solve. The lack of inter-element communication and easily parallelizable nature of the local matrix generation stage coupled with the parallelization techniques developed for the linear system solvers make a numerical scheme for elliptic PDEs a good candidate for implementation on streaming architectures such as modern graphical processing units (GPUs). We propose an algorithmic pipeline for mapping an elliptic finite element method to the GPU and perform a case study for a particular method within the HDG framework. This study provides comparison between CPU and GPU implementations of the method as well as highlights certain performance-crucial implementation details. The choice of the HDG method for the case study was dictated by the computationally-heavy local matrix generation stage as well as the reduced trace-based communication pattern, which together make the method amenable to the fine-grained parallelism of GPUs. We demonstrate that the HDG method is well-suited for GPU implementation, obtaining total speedups on the order of 30–35 times over a serial CPU implementation for moderately sized problems.     

Computational error-analysis of a discontinuous Galerkin discretization applied to large-eddy simulation of homogeneous turbulence

A computational error-assessment of large-eddy simulation (LES) in combination with a discontinuous Galerkin finite element method is presented for homogeneous, isotropic, decaying turbulence. The error-landscape database approach is used to quantify the total simulation error that arises from the use of the Smagorinsky eddy-viscosity model in combination with the Galerkin discretization. We adopt a modified HLLC flux, allowing an explicit control over the dissipative component of the numerical flux. The optimal dependence of the Smagorinsky parameter on the spatial resolution is determined for second and third order accurate Galerkin methods. In particular, the role of the numerical dissipation relative to the contribution from the Smagorinsky dissipation is investigated. We observed an ‘exchange of dissipation’ principle in the sense that an increased numerical dissipation implied a reduction in the optimal Smagorinsky parameter. The predictions based on Galerkin discretization with fully stabilized HLLC flux were found to be less accurate than when a central discretization with (mainly) Smagorinsky dissipation was used. This was observed for both the second and third order Galerkin discretization, suggesting to emphasize central discretization of the convective nonlinearity and stabilization that mimics eddy-viscosity as sub-filter dissipation.     

A Numerical Comparison of the Lax–Wendroff Discontinuous Galerkin Method Based on Different Numerical Fluxes

Discontinuous Galerkin (DG) method is a spatial discretization procedure, employing useful features from high-resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters. In [(2005). Comput. Methods Appl. Mech. Eng. 194, 4528], we developed a Lax–Wendroff time discretization procedure for the DG method (LWDG), an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. In most of the DG papers in the literature, the Lax–Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes, which could also be used. In this paper, we systematically investigate the performance of the LWDG method based on different numerical fluxes, including the first-order monotone fluxes such as the Godunov flux, the Engquist–Osher flux, etc., the second-order TVD fluxes and generalized Riemann solver, with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one-dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two-dimensional systems.      

Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

This paper proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients, under the finite noise assumption. First, the stochastic discontinuous Galerkin method represents the stochastic solution in a Galerkin framework. Second, the Monte Carlo discontinuous Galerkin method samples the coefficients by a Monte Carlo approach. Both methods discretize the differential operators by the class of interior penalty discontinuous Galerkin methods. Error analysis is obtained. Numerical results show the sensitivity of the expected value and variance with respect to the penalty parameter of the spatial discretization.     

基于Matlab的SMB色谱分离过程计算机仿真研究

针对考虑因素全面的SMB综合速率模型，采用有限元方法和正交配点法分别对柱向和吸附剂颗粒径向模型进行离散化，利用Matlab ODE求解器对SMB过程进行了数值求解，并编制了SMB过程仿真软件。在此基础上进行了一个SMB分离实例仿真，分析了切换时间和流量变化对分离性能的影响，验证了SMB过程分离性能的参数敏感性，提出了对系统实施先进控制的必要性。     

Discontinuous Hamiltonian Finite Element Method for Linear Hyperbolic Systems

We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, “mass” and energy are preserved. We discretize the bilinear Poisson bracket in each element with discontinuous elements and introduce numerical fluxes via integration by parts while preserving the skew-symmetry of the bracket. This automatically results in a mass and energy conservative discretization. When combined with a symplectic time integration method, energy is approximately conserved and shows no drift. For comparison, the discontinuous Galerkin method for this problem is also used. A variety numerical examples is shown to illustrate the accuracy and capability of the new method.     

Discontinuous Galerkin solution of compressible flow in time-dependent domains

This work is concerned with the simulation of inviscid compressible flow in time-dependent domains. We present an arbitrary Lagrangian–Eulerian (ALE) formulation of the Euler equations describing compressible flow, discretize them in space by the discontinous Galerkin method and introduce a semi-implicit linearized time stepping for the numerical solution of the complete problem. Special attention is paid to the treatment of boundary conditions and the limiting procedure avoiding the Gibbs phenomenon in the vicinity of discontinuities. The presented computational results show the applicability of the developed method.     

Use of local grid refinement and a Galerkin technique to study secondary migration in fractured and faulted regions

A simulator, which includes a full three dimensional, multi-component and three phase model for secondary oil migration is used to study flow through fractured and faulted regions. A control volume discretization is used to discretize the strongly coupled system of partial differential equations the model creates. Non-regular and non-matching local grid refinement (LGR) is used in critical areas to ensure the required accuracy. Two solution methods for solving the composite problem LGR create are studied. The simplest one uses the GMRES solver directly on the composite problem. The second method splits the composite problem into one coarse problem and one problem for each refined area. This method is a two level multigrid method which is based on a Galerkin technique. The CPU time and the number of iteration needed for convergence in both methods are presented and compared. Received: 23 February 1999 / Accepted: 17 June 1999     

An HDG Method with Orthogonal Projections in Facet Integrals

We propose and analyze a new hybridizable discontinuous Galerkin (HDG) method for second-order elliptic problems. Our method is obtained by inserting the \(L^2\)-orthogonal projection onto the approximate space for a numerical trace into all facet integrals in the usual HDG formulation. The orders of convergence for all variables are optimal if we use polynomials of degree \(k+l\), \(k+1\) and k, where k and l are any non-negative integers, to approximate the vector, scalar and trace variables, which implies that our method can achieve superconvergence for the scalar variable without postprocessing. Numerical results are presented to verify the theoretical results.     

A Comparison of HDG Methods for Stokes Flow

In this paper, we compare hybridizable discontinuous Galerkin (HDG) methods for numerically solving the velocity-pressure-gradient, velocity-pressure-stress, and velocity-pressure-vorticity formulations of Stokes flow. Although they are defined by using different formulations of the Stokes equations, the methods share several common features. First, they use polynomials of degree k for all the components of the approximate solution. Second, they have the same globally coupled variables, namely, the approximate trace of the velocity on the faces and the mean of the pressure on the elements. Third, they give rise to a matrix system of the same size, sparsity structure and similar condition number. As a result, they have the same computational complexity and storage requirement. And fourth, they can provide, by means of an element-by element postprocessing, a new approximation of the velocity which, unlike the original velocity, is divergence-free and H(div)-conforming. We present numerical results showing that each of the approximations provided by these three methods converge with the optimal order of k+1 in L ² for any k≥0. We also display experiments indicating that the postprocessed velocity is a better approximation than the original approximate velocity. It converges with an additional order than the original velocity for the gradient-based HDG, and with the same order for the vorticity-based HDG methods. For the stress-based HDG methods, it seems to converge with an additional order for even polynomial degree approximations. Finally, the numerical results indicate that the method based on the velocity-pressure-gradient formulation provides the best approximations for similar computational complexity.