The WKB Local Discontinuous Galerkin Method for the Simulation of Schrödinger Equation in a Resonant Tunneling Diode

In this paper, we develop a multiscale local discontinuous Galerkin (LDG) method to simulate the one-dimensional stationary Schrödinger-Poisson problem. The stationary Schrödinger equation is discretized by the WKB local discontinuous Galerkin (WKB-LDG) method, and the Poisson potential equation is discretized by the minimal dissipation LDG (MD-LDG) method. The WKB-LDG method we propose provides a significant reduction of both the computational cost and memory in solving the Schrödinger equation. Compared with traditional continuous finite element Galerkin methodology, the WKB-LDG method has the advantages of the DG methods including their flexibility in h-p adaptivity and allowance of complete discontinuity at element interfaces. Although not addressed in this paper, a major advantage of the WKB-LDG method is its feasibility for two-dimensional devices.     

Numerical modeling of air mass motion by the mesh-free method

The Navier-Stokes equation, applied to the calculation of wind velocity without accounting for the turbulent motion of the atmosphere, is considered in this work. The main flow characteristics were computed with the use of the Lagrange discrete vortex method for finding the solution of the Poisson equation under the Dirichlet and Neumann boundary conditions. To do this, two mesh-free methods: the element-free Galerkin (EFG) and the Finite Pointset (FP) methods, as well as the modification of the latter, have been analyzed. It is shown that the computation speed of the EFG method is higher than of the FP-method. It is determined that a serious disadvantage of the FP-method is its low rate convergence, while the computational complexity of each iteration is reasonable. The use of the modified FP-method has shown its computational speed to be comparable with that of the EFG method, although the advantage of the FP-method is not obvious when the size of the problem increases.     

势问题的复变量无单元Galerkin方法

为提高无单元Galerkin(Element-Free Galerkin, EFG)方法的计算效率,将复变量移动最小二乘法与EFG方法结合,利用控制方程的积分弱形式并采用Lagrange乘子法引入边界条件,提出势问题的复变量无单元Galerkin(Complex Variable EFG,CVEFG)方法,并推导相关公式.与传统的EFG方法相比,该方法采用复变量移动最小二乘法可以减少试函数中的待定系数,从而减少计算量、提高计算效率. 最后,给出数值算例验证该方法的有效性.     

A Fully Discrete Fast Fourier–Galerkin Method Solving a Boundary Integral Equation for the Biharmonic Equation

We develop a fully discrete fast Fourier–Galerkin method for solving a boundary integral equation for the biharmonic equation by introducing a quadrature scheme for computing the integrals of non-smooth functions that appear in the Fourier–Galerkin method. A key step in developing the fully discrete fast Fourier–Galerkin method is the design of a fast quadrature scheme for computing the Fourier coefficients of the non-smooth kernel function involved in the boundary integral equation. We prove that with the proposed quadrature algorithm, the total number of additions and multiplications used in generating the compressed coefficient matrix for the proposed method is quasi-linear (linear with a logarithmic factor), and the resulting numerical solution of the equation preserves the optimal convergence order. Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed method.     

Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations

In this paper, we continue our investigation of the locally divergence-free discontinuous Galerkin method, originally developed for the linear Maxwell equations (J. Comput. Phys. 194 588–610 (2004)), to solve the nonlinear ideal magnetohydrodynamics (MHD) equations. The distinctive feature of such method is the use of approximate solutions that are exactly divergence-free inside each element for the magnetic field. As a consequence, this method has a smaller computational cost than the traditional discontinuous Galerkin method with standard piecewise polynomial spaces. We formulate the locally divergence-free discontinuous Galerkin method for the MHD equations and perform extensive one and two-dimensional numerical experiments for both smooth solutions and solutions with discontinuities. Our computational results demonstrate that the locally divergence-free discontinuous Galerkin method, with a reduced cost comparing to the traditional discontinuous Galerkin method, can maintain the same accuracy for smooth solutions and can enhance the numerical stability of the scheme and reduce certain nonphysical features in some of the test cases.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Error Analysis of Chebyshev-Legendre Pseudo-spectral Method for a Class of Nonclassical Parabolic Equation

Many physical phenomena are modeled by nonclassical parabolic initial boundary value problems which involve a nonclassical term u _xxt in the governed equation. Combining with the Crank-Nicolson/leapfrog scheme in time discretization, Chebyshev-Legendre pseudo-spectral method is applied to space discretization for numerically solving the nonclassical parabolic equation. The proposed approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the proposed method, the computational complexity is reduced and both accuracy and efficiency are achieved. The stability and convergence are rigorously set up. The convergence rate shows ??spectral accuracy??. Numerical experiments are presented to demonstrate the effectiveness of the method and to confirm the theoretical results.     

A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems

We give a space-time Galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree p>0 (cG(p)), with a discontinuous Galerkin piecewise constant (dG(0)) or linear (dG(1)) approximation in time. A posteriori Galerkin-error estimates are derived by exploiting the Galerkin framework and optimal stability estimates for a related dual backward problem. The a posteriori error estimates are quite flexible: strong L_p-energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth, while weak-energy norms are used when the data are non-smooth (in time).We also give upper bounds on the dG(0)cG(1) a posteriori error estimates which indicate optimality. However, a complete analysis is not given.     

A novel reduced-basis method with upper and lower bounds for real-time computation of linear elasticity problems

This paper presents a novel reduced-basis method for analyzing problems of linear elasticity in a systematical, rapid and reliable fashion for solutions with both upper and lower bounds to the exact solution in the form of energy norm or compliance output. The lower bound of the solution output is obtained form the well-known reduced-basis method based on the Galerkin projection used in the finite element method, which is termed as GP_RBM. For the upper bound, a new reduced-basis approach is developed by the combination of the reduced-basis method and a smoothed Galerkin projection used in the linearly conforming point interpolation method, and it is thus termed as SGP_RBM. To examine the present SGP_RBM, we first conduct a theoretical study on the very important upper bound property. Reduced-basis models for both GP_RBM and SGP_RBM are constructed with the aid of an asymptotic error estimation and greedy adaptive procedure. The GP_RBM and the newly proposed SGP_RBM are applied to analyze a cantilever beam with an oblique crack to verify the proposed RBM technique in terms of accuracy, convergence, bound properties and computational savings. Both theoretical analysis and numerical results have demonstrated that the present method is a very efficient method for real-time solutions providing exact output bounds.     

Local projection stabilisation on S-type meshes for convection–diffusion problems with characteristic layers

Singularly perturbed convection–diffusion problems with exponential and characteristic layers are considered on the unit square. The discretisation is based on layer-adapted meshes. The standard Galerkin method and the local projection scheme are analysed for bilinear and higher order finite element where enriched spaces were used. For bilinears, first order convergence in the ε-weighted energy norm is shown for both the Galerkin and the stabilised scheme. However, supercloseness results of second order hold for the Galerkin method in the ε-weighted energy norm and for the local projection scheme in the corresponding norm. For the enriched ${\mathcal{Q}_p}$ -elements, p ≥ 2, which already contain the space ${\mathcal{P}_{p+1}}$ , a convergence order p + 1 in the ε-weighted energy norm is proved for both the Galerkin method and the local projection scheme. Furthermore, the local projection methods provides a supercloseness result of order p + 1 in local projection norm.     

BDDC preconditioning for high-order Galerkin Least-Squares methods using inexact solvers

A high-order Galerkin Least-Squares (GLS) finite element discretization is combined with a Balancing Domain Decomposition by Constraints (BDDC) preconditioner and inexact local solvers to provide an efficient solution technique for large-scale, convection-dominated problems. The algorithm is applied to the linear system arising from the discretization of the two-dimensional advection–diffusion equation and Euler equations for compressible, inviscid flow. A Robin–Robin interface condition is extended to the Euler equations using entropy-symmetrized variables. The BDDC method maintains scalability for the high-order discretization of the diffusion-dominated flows, and achieves low iteration count in the advection-dominated regime. The BDDC method based on inexact local solvers with incomplete factorization and p = 1 coarse correction maintains the performance of the exact counterpart for the wide range of the Peclet numbers considered while at significantly reduced memory and computational costs.     

Numerical analysis of fluid squeezed between two parallel plates by meshless method

This paper presents the steady-state and transient analysis of the fluid squeezed between two long parallel plates. The governing coupled partial differential equations have been discretized by element free Galerkin method and implemented using variational approach. Penalty and Lagrange multiplier techniques have been utilized to enforce the essential boundary conditions. Four point Gauss quadrature has been used to evaluate the viscous terms in the coefficient matrix whereas reduced integration scheme (i.e. one point Gauss quadrature) has been used to evaluate the penalty terms over two-dimensional domain (Ω). Cubicspline, exponential and rational weight functions have been used in the present work. The results obtained by EFG method are compared with those obtained by finite element and analytical methods. The effect of scaling and penalty parameters on EFG results has been discussed in detail.     

接触力学自适应无网格计算系统设计

在求解弹塑性接触问题时,为了保证计算精度同时减少计算耗费,提出一种接触力学自适应无网格计算系统.利用无网格法结点排布灵活便于删减等优点,对采用基于应变能梯度的自适应无网格伽辽金一有限元方法求解接触问题的计算系统的原理、算法流程进行阐述,并考虑到自适应影响域半径、弹塑性等关键问题.采用模块化思想进行程序设计.通过圆柱体与弹塑性平面接触的算例验证了计算系统的正确性,并将自适应密化解与整体密化解比较,显现出良好的计算精度和效率.为接触力学自适应无网格分析软件商业化提供了一种理论和程序设计基础.     

Positive solutions for a class of elliptic equations

In this paper, the existence and bifurcation of positive solutions for a class of elliptic equations involving the p-Laplacian are studied by using the sub- and super-solution method and the topological degree argument.     

On the Negative-Order Norm Accuracy of a Local-Structure-Preserving LDG Method

The accuracy in negative-order norms is examined for a local-structure-preserving local discontinuous Galerkin method for the Laplace equation (Li and Shu, in Methods Appl. Anal. 13:215–233, 2006). With its distinctive feature in using harmonic polynomials as local approximating functions, this method has lower computational complexity than the standard local discontinuous Galerkin method while keeping the same order of accuracy in both the energy and the L ² norms. In this note, numerical experiments are presented to demonstrate some accuracy loss of the method in negative-order norms.     

Adaptive element-free Galerkin method applied to the limit analysis of plates

The implementation of an h-adaptive element-free Galerkin (EFG) method in the framework of limit analysis is described. The naturally conforming property of meshfree approximations (with no nodal connectivity required) facilitates the implementation of h-adaptivity. Nodes may be moved, discarded or introduced without the need for complex manipulation of the data structures involved. With the use of the Taylor expansion technique, the error in the computed displacement field and its derivatives can be estimated throughout the problem domain with high accuracy. A stabilized conforming nodal integration scheme is extended for use in error estimation and results in an efficient and truly meshfree adaptive method. To demonstrate its effectiveness the procedure is then applied to plates with various boundary conditions.     

A posteriori error estimates for fem–bem couplings of three-dimensional electromagnetic problems

We present residual based and p-hierarchical a posteriori error estimators for a Galerkin method coupling finite elements and boundary elements for time–harmonic interface problems in electromagnetics; special emphasis is taken for the eddy current problem. The Galerkin discretization uses lowest order Nédélec elements in the interior domain and vectorial surface rotations of continuous, piecewise linear functions on the interface boundary. Singular, weakly singular and hypersingular boundary integral operators appearing in the variational formulation show up in the terms of the error estimators as well. The estimators are derived from the defect equation using Helmholtz and Hodge decompositions. Numerical tests underline reliability and efficiency of the given error estimators yielding reasonable mesh refinements.     

Hybrid numerical methods for convection-diffusion problems in arbitrary geometries

The hybrid nodal-integral/finite element method (NI-FEM) and the hybrid nodal-integral/finite analytic method (NI-FAM) are developed to solve the steady-state, two-dimensional convection-diffusion equation (CDE). The hybrid NI-FAM for the steady-state problem is then extended to solve the more general time-dependent, two-dimensional, CDE. These hybrid coarse mesh methods, unlike the conventional nodal-integral approach, are applicable in arbitrary geometries and maintain the high efficiency of the conventional nodal-integral method (NIM). In steady-state problems, the computational domain for both hybrid methods is discretized using rectangular nodes in the interior of the domain and along vertical and horizontal boundaries, while triangular nodes are used along the boundaries that are not parallel to the x or y axes. In time-dependent problems, the rectangular and triangular nodes become space-time parallelepiped and wedge-shaped nodes, respectively. The difference schemes for the variables on the interfaces of adjacent rectangular/parallelepiped nodes are developed using the conventional NIM. For the triangular nodes in the hybrid NI-FEM, a trial function is written in terms of the edge-averaged concentration of the three edges and made to satisfy the CDE in an integral sense. In the hybrid NI-FAM, the concentration over the triangular/wedge-shaped nodes is represented using a finite analytic approximation, which is based on the analytic solution of the one-dimensional CDE. The difference schemes for both hybrid methods are then developed for the interfaces between the rectangular/parallelepiped and triangular/wedge-shaped nodes by imposing continuity of the flux across the interfaces. A formal derivation of these hybrid methods and numerical results for several test problems are presented and discussed.     

Implementation of a discontinuous Galerkin morphological model on two-dimensional unstructured meshes

The shallow water equations are used to model large-scale surface flow in the ocean, coastal rivers, estuaries, salt marshes, bays, and channels. They can describe tidal flows as well as storm surges associated with extreme storm events, such as hurricanes. The resulting currents can transport bed load and suspended sediment and result in morphological changes to the seabed. Modeling these processes requires tightly coupling a bed morphology equation to the shallow water equations. Discontinuous Galerkin finite element methods are a natural choice for modeling this coupled system, given the need to solve these problems on unstructured computational meshes, as well as the desire to implement hp-adaptivity for capturing the dynamic features of the solution. However, because of the presence of non-conservative products in the momentum equations, the standard DG method cannot be applied in a straightforward manner. To rectify this situation, we summarize and follow an extended approach described by Rhebergen et al., which uses theoretical results due to Dal Maso et al. appearing in earlier work. In this paper, we focus on aspects of the implementation of the morphological model for bed evolution within the Advanced Circulation (ADCIRC) modeling framework, as well as the verification of the RKDG method in both h (mesh spacing) and p (polynomial order). This morphological model is applied to a number of coastal engineering problems, and numerical results are presented, with attention paid to the effects of h- and p-refinement in these applications. In particular, it is observed that for sediment transport, piecewise constant (i.e., finite volume) approximations of the bed are very over-diffusive and lead to poor sediment solutions.     

A framework combining meshfree analysis and adaptive kriging for optimization of stiffened panels

Structural and Multidisciplinary Optimization - A framework is developed for structural optimization using an Element Free Galerkin (EFG) method for analyzing the structure, a kriging for surrogate...