The Compact Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity

A compact discontinuous Galerkin method (CDG) is devised for nearly incompressible linear elasticity, through replacing the global lifting operator for determining the numerical trace of stress tensor in a local discontinuous Galerkin method (cf. Chen et al., Math Probl Eng 20, 2010) by the local lifting operator and removing some jumping terms. It possesses the compact stencil, that means the degrees of freedom in one element are only connected to those in the immediate neighboring elements. Optimal error estimates in broken energy norm, $H^1$ -norm and $L^2$ -norm are derived for the method, which are uniform with respect to the Lamé constant $\lambda .$ Furthermore, we obtain a post-processed $H(\text{ div})$ -conforming displacement by projecting the displacement and corresponding trace of the CDG method into the Raviart–Thomas element space, and obtain optimal error estimates of this numerical solution in $H(\text{ div})$ -seminorm and $L^2$ -norm, which are uniform with respect to $\lambda .$ A series of numerical results are offered to illustrate the numerical performance of our method.     

Numerical Simulation for Porous Medium Equation by Local Discontinuous Galerkin Finite Element Method

In this paper we will consider the simulation of the local discontinuous Galerkin (LDG) finite element method for the porous medium equation (PME), where we present an additional nonnegativity preserving limiter to satisfy the physical nature of the PME. We also prove for the discontinuous ℙ₀ finite element that the average in each cell of the LDG solution for the PME maintains nonnegativity if the initial solution is nonnegative within some restriction for the flux’s parameter. Finally, numerical results are given to show the advantage of the LDG method for the simulation of the PME, in its capability to capture accurately sharp interfaces without oscillation. The research of Q. Zhang is supported by CNNSF grant 10301016.     

A Discontinuous Galerkin Method for the Subjective Surfaces Problem

The work formulates and evaluates the local discontinuous Galerkin method for the subjective surfaces problem based on the curvature driven level set equation. A new mixed formulation simplifying the treatment of nonlinearities is proposed. The numerical algorithm is evaluated using several artificial and realistic test cases.     

Local Analysis of the Local Discontinuous Galerkin Method with Generalized Alternating Numerical Flux for One-Dimensional Singularly Perturbed Problem

In this paper we would like to present the local analysis of the local discontinuous Galerkin method based on the generalized alternating numerical flux for the one-dimensional time-dependent singularly perturbed problem with a stationary boundary layer. By virtue of the generalized Gauss–Radau projection and energy technique with suitable weight function, we can obtain the double-optimal local error estimate that the convergence rate in \(\hbox {L}^2\)-norm out of the pollution region nearby the outflow boundary is optimal, and the width of pollution region is quasi-optimal also. Numerical experiments are given.     

A Fully-Discrete Local Discontinuous Galerkin Method for Convection-Dominated Sobolev Equation

In this paper we shall present, for the convection-dominated Sobolev equations, the fully-discrete numerical scheme based on the local discontinuous Galerkin (LDG) finite element method and the third-order explicitly total variation diminishing Runge-Kutta (TVDRK3) time marching. A priori error estimate is obtained for any piecewise polynomials of degree at most k≥1, under the general spatial-temporal restriction. The bounded constant in error estimate is independent of the reciprocal of the diffusion and dispersion coefficients, after removing the effect of smoothness of the exact solution. Finally some numerical results are given to verify the presented conclusion.     

A Discontinuous Galerkin Method for Linear Symmetric Hyperbolic Systems in Inhomogeneous Media

The Discontinuous Galerkin (DG) method provides a powerful tool for approximating hyperbolic problems. Here we derive a new space-time DG method for linear time dependent hyperbolic problems written as a symmetric system (including the wave equation and Maxwell’s equations). The main features of the scheme are that it can handle inhomogeneous media, and can be time-stepped by solving a sequence of small linear systems resulting from applying the method on small collections of space-time elements. We show that the method is stable provided the space-time grid is appropriately constructed (this corresponds to the usual time-step restriction for explicit methods, but applied locally) and give an error analysis of the scheme. We also provide some simple numerical tests of the algorithm applied to the wave equation in two space dimensions (plus time).This revised version was published online in July 2005 with corrected volume and issue numbers.     

An hp-Analysis of the Local Discontinuous Galerkin Method for Diffusion Problems

We present an hp-error analysis of the local discontinuous Galerkin method for diffusion problems, considering unstructured meshes with hanging nodes and two- and three-dimensional domains. Our estimates are optimal in the meshsize h and slightly suboptimal in the polynomial approximation order p. Optimality in p is achieved for matching grids and polynomial boundary conditions.     

An A Posteriori Error Estimate for the Local Discontinuous Galerkin Method

In this work an a posteriori global error estimate for the Local Discontinuous Galerkin (LDG) applied to a linear second order elliptic problem is analyzed. Using a mixed formulation, an upper bound of the error in the primal variable is derived from explicit computations. Finally, a local adaptive scheme based on explicit error estimators is studied numerically using one dimensional problems.This revised version was published online in July 2005 with corrected volume and issue numbers.     

A Simple Preconditioner for a Discontinuous Galerkin Method for the Stokes Problem

In this paper we construct Discontinuous Galerkin approximations of the Stokes problem where the velocity field is $H(\mathrm{div},\Omega )$ -conforming. This implies that the velocity solution is divergence-free in the whole domain. This property can be exploited to design a simple and effective preconditioner for the final linear system.     

Local Discontinuous Galerkin Method for Surface Diffusion and Willmore Flow of Graphs

In this paper, we develop a local discontinuous Galerkin (LDG) finite element method for surface diffusion and Willmore flow of graphs. We prove L ² stability for the equation of surface diffusion of graphs and energy stability for the equation of Willmore flow of graphs. We provide numerical simulation results for different types of solutions of these two types of the equations to illustrate the accuracy and capability of the LDG method.     

Discontinuous Galerkin Method for Investigating Ice Strength

This paper discusses the numerical modeling of various ice-strength measurement experiments, including uniaxial compression and bending, and it also compares the data obtained by field and numerical experiments. Numerical simulation is based on a dynamic system of continuum mechanics equations with ice considered as an elasto-plastic medium with brittle and crushing fracture dynamic criteria. The simulation software developed by the authors is based on the discontinuous Galerkin method and runs on high-performance systems with a distributed memory. Estimating the explicit values used by the mathematical models poses a major problem because some of them cannot be directly measured in field experiments due to the multiple interferences of physical processes. In practice, it is only possible to directly measure their total influence. However, this problem can be solved by comparing the numerical experiment with the field data. As a result of this work, the elasto-plastic ice model is verified and some missing physical properties are obtained by the numerical experiments.     

A Local Discontinuous Galerkin Method for the Propagation of Phase Transition in Solids and Fluids

A local discontinuous Galerkin (LDG) finite element method for the solution of a hyperbolic–elliptic system modeling the propagation of phase transition in solids and fluids is presented. Viscosity and capillarity terms are added to select the physically relevant solution. The $L^2-$ stability of the LDG method is proven for basis functions of arbitrary polynomial order. In addition, using a priori error analysis, we provide an error estimate for the LDG discretization of the phase transition model when the stress–strain relation is linear, assuming that the solution is sufficiently smooth and the system is hyperbolic. Also, results of a linear stability analysis to determine the time step are presented. To obtain a reference exact solution we solved a Riemann problem for a trilinear strain–stress relation using a kinetic relation to select the unique admissible solution. This exact solution contains both shocks and phase transitions. The LDG method is demonstrated by computing several model problems representing phase transition in solids and in fluids with a Van der Waals equation of state. The results show the convergence properties of the LDG method.     

A Coupling of Local Discontinuous Galerkin and Natural Boundary Element Method for Exterior Problems

In this paper, we apply the coupling of local discontinuous Galerkin (LDG) and natural boundary element method(NBEM) to solve a two-dimensional exterior problem. As a consequence, the main features of LDG and NBEM are maintained and hence the coupled approach benefits from the advantages of both methods. Referring to Gatica et al. (Math. Comput. 79(271):1369?C1394, 2010), we employ LDG subspaces whose functions are continuous on the coupling boundary. In this way, the primitive variables become the only boundary unknown, and hence the total number of unknown functions is reduced. We prove the stability of the new discrete scheme and derive an a priori error estimate in the energy norm. Some numerical examples conforming the theoretical results are provided.     

Discontinuous Galerkin Method for Linear Free-Surface Gravity Waves

In this paper, we discuss a discontinuous Galerkin finite (DG) element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the L²-norm is in both cases of optimal order and proportional to O(Δt²+h^p+1), without the need for a separate velocity reconstruction, with p the polynomial order, h the mesh size and Δt the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.This revised version was published online in July 2005 with corrected volume and issue numbers.     

An Analysis of the Minimal Dissipation Local Discontinuous Galerkin Method for Convection–Diffusion Problems

We analyze the so-called the minimal dissipation local discontinuous Galerkin method (MD-LDG) for convection–diffusion or diffusion problems. The distinctive feature of this method is that the stabilization parameters associated with the numerical trace of the flux are identically equal to zero in the interior of the domain; this is why its dissipation is said to be minimal. We show that the orders of convergence of the approximations for the potential and the flux using polynomials of degree k are the same as those of all known discontinuous Galerkin methods, namely, (k + 1) and k, respectively. Our numerical results verify that these orders of convergence are sharp. The novelty of the analysis is that it bypasses a seemingly indispensable condition, namely, the positivity of the above mentioned stabilization parameters, by using a new, carefully defined projection tailored to the very definition of the numerical traces.     

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.     

Unstructured Grid-Based Discontinuous Galerkin Method for Broadband Electromagnetic Simulations

A parallel, unstructured, high-order discontinuous Galerkin method is developed for the time-dependent Maxwell's equations, using simple monomial polynomials for spatial discretization and a fourth-order Runge–Kutta scheme for time marching. Scattering results for a number of validation cases are computed employing polynomials of up to third order. Accurate solutions are obtained on coarse meshes and grid convergence is achieved, demonstrating the capabilities of the scheme for time-domain electromagnetic wave scattering simulations.