Structural joining method for the solution of the model Lighthill equation with a regular singular point

Using the structural joining method, we construct a uniformly valid explicit asymptotics of the solution of a perturbed model Lighthill equation with a regular singular point.     

A posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems

In this paper, we consider the a posteriori error analysis of discontinuous Galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions. We establish several residual-based a posteriori error estimators which provide global upper bounds and a local lower bound on the error. Further, for nonsteady problem, we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space. Our a posteriori error analysis is based on the mesh-dependent a priori estimates for the first order hyperbolic problems. These a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods.     

Numerical methods and error analysis for one dimensional elliptic problems containing singularities

The method of auxiliary mapping (MAM), introduced by Babu?ka and Oh, was proven to be very successful in dealing with monotone singularities arising in two‐dimensional problems. In this article, in the framework of the p‐version of FEM, MAM is presented for one‐dimensional elliptic boundary value problems containing singularities. Moreover, in order to show the effectiveness of MAM, a detailed proof of an error estimate is also presented, which gives a sharp error bound of MAM. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 399–420, 2003.     

On error estimates in the Galerkin method for hyperbolic equations

We consider the Cauchy problem in a Hilbert space for a second-order abstract quasilinear hyperbolic equation with variable operator coefficients and nonsmooth (but Bochner integrable) free term. For this problem, we establish an a priori energy error estimate for the semidiscrete Galerkin method with an arbitrary choice of projection subspaces. Also, we establish some results on existence and uniqueness of an exact weak solution. We give an explicit error estimate for the finite element method and the Galerkin method in Mikhlin form.     

Numerical simulations of the improved Boussinesq equation

In this study, numerical simulations of the improved Boussinesq equation are obtained using two finite difference schemes and two finite element methods, based on the second‐and third‐order time discretization. The methods are tested on the problems of propagation of a soliton and interaction of two solitons. After the L_∞ error norm is used to measure differences between the exact and numerical solutions, the results obtained by the proposed methods are compared with recently published results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

非线性双曲型方程的混合元方法

1　引　　言考虑下述非线性双曲型方程的混合问题:c(x,u)utt-.(a(x,u)u)=f(x,u,t),　　x∈Ω,t∈J,(1.1)u(x,0)=u0(x),　　x∈Ω,(1.2)ut(x,0)=u1(x),　　x∈Ω,(1.3)u(x,t)=-g(x,t),　　(x,t)∈Ω×J,(1.4)其中ΩR2是一具有Lipschitz边界Ω的有界区域,J=[0,T],0相似文献   

Developing weak Galerkin finite element methods for the wave equation

In this article, we extend the recently developed weak Galerkin method to solve the second‐order hyperbolic wave equation. Many nice features of the weak Galerkin method have been demonstrated for elliptic, parabolic, and a few other model problems. This is the initial exploration of the weak Galerkin method for solving the wave equation. Here we successfully developed and established the stability and convergence analysis for the weak Galerkin method for solving the wave equation. Numerical experiments further support the theoretical analysis. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 868–884, 2017     

Numerical results for the Klein‐Gordon equation in de Sitter spacetime

We consider the initial value problem for the Klein‐Gordon equation in de Sitter spacetime. We use the central difference scheme on the temporal discretization. We also discretize the spatial variable using the finite element method with implicit and the Crank‐Nicolson schemes for the numerical solution of the initial value problem. In order to show the accuracy for the results of the solutions, we also examine the finite difference methods. We observe that the numerical results obtained by using these methods are compatible.     

Guaranteed velocity error control for the pseudostress approximation of the Stokes equations

The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in L². Any standard mixed finite element function space can be utilized for this mixed formulation, e.g., the Raviart‐Thomas discretization which is related to the Crouzeix‐Raviart nonconforming finite element scheme in the lowest‐order case. The effective and guaranteed a posteriori error control for this nonconforming velocity‐oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf‐sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1411–1432, 2016     

一维常系数双曲型方程每步过滤解法的稳定性与误差分析

考虑一维常系数双曲型方程的高阶数值解法，引进了一种稳定性过滤算子FN，最后从理论和数值实验上分析了该过滤算子对数值解稳定性和精度的影响．     

广义热传导方程有限元算法的计算准则

本文作者曾对经典的(抛物型)热传导方程提出了两种单调性的新概念,推导并证明了几组计算准则,可以使其有限元数值解消除很容易出现的振荡和超界现象.本文把上述成果用于广义(双曲型)热传导方程的有限元解中,推导出它的有限元解的计算准则,并获得了一些新结论.     

Superconvergence of Finite Element Approximations to Parabolic and Hyperbolic Integro-Differential Equations

1 IntroductionRecently, considerable attention has been devoted to the finite element analysis fOr par-tiaJ illtegro-differential equations, see, fOr example, Yanik and Ftweatherl1], Cannon andLin[2'3] 5 Chen and Shih[4], Lin, Thomee and Wahlbi.I5l ? Thomee and Zhang[6] and ZhangI7j8J.The main tool used for this kind of equations is the mtz-Voterra projection [5'7] as againstttitz projection fOr parabolic equation. In this paPer, we are concerned primarily with theanalysis of knot supe…     

Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method

In this article we propose a numerical scheme to solve the one‐dimensional hyperbolic telegraph equation. The method consists of expanding the required approximate solution as the elements of shifted Chebyshev polynomials. Using the operational matrices of integral and derivative, we reduce the problem to a set of linear algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Numerical analysis of the hyperbolic two-temperature model

The system of hyperbolic heat conduction problems is solved numerically. The explicit and fully implicit Euler type schemes for the time integration of the nonstationary problem are proposed and investigated. Space derivatives are approximated by using the finite volume method, resulting in conservative and monotonous discrete approximations of the second order of accuracy. The stability analysis is done in the L ₂ and energy norms for a simplified one-temperature equation and the system of two equations, describing the temperature and the flux. Results of numerical experiments are presented. This work was supported by the Lithuanian State Science and Studies Foundation within the projects B-03/2007, B-09/2007 and by the Agency for International Science and Technology Development Programmes in Lithuania within the EUREKA projects E!3691 OPTCABLES and E!3483 EULASNET LASCAN.     

On the use of physical spline finite element method for acoustic scattering

This paper presents the comparison of physical spline finite element method (PSFEM), in which differential equations are incorporated into interpolations of basic elements, with least-squares finite element method (LSFEM) and mixed Galerkin finite element method (MGFEM) on the numerical solution of one dimensional Helmholtz equation applied to an acoustic scattering problem. Firstly, all three methods are explained in detail and then it is shown that PSFEM reaches higher precision in a shorter time with fewer nodes than the other methods. It is also observed that this method is well suited for high frequency acoustic problems. Consequently, the results of PSFEM point out better efficiency in terms of number of unknowns and accuracy level.     

Numerical simulation of the wave equation with discontinuous coefficients by nonconforming finite elements

The goal of this article is to apply the mortar finite element method to the numerical simulation of (electromagnetic and/or acoustic) waves propagating in an inhomogeneous support. This approach allows us to use meshes well adapted to the local physical parameters of the media without any conformity constraints. A complete mathematical study is supplied providing the expected optimal convergence rate. Numerical performances of such a technique, as well as its advantages, are also discussed. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 637–656, 1999     

Global error bounds for the Petrov–Galerkin discretization of the neutron transport equation

In this paper, we prove that the piecewise bilinear Petrov‐Galerkin discretization for the mono‐directional neutron transport equation described in (J. Comput. Phys. 1986; 64 :96–111) is convergent and second‐order accurate, provided that the true solution to the problem has continuous partial derivatives of all orders up through three. We do this by giving a bound on the 2‐norm of the inverse of the system matrix that is independent of the mesh size. This shows that the global error is of the same order as the local truncation error. Copyright © 2010 John Wiley & Sons, Ltd.     

带损伤弹性反问题的数值分析

考虑一类由椭圆性方程和热传导方程共同来刻画的准静态弹性模型,通过给定观测值来反演边界的牵引力.首先构造一个凸目标泛函,并引入Tikhonov正则化方法,使之极小化得到一个稳定的近似解.再用有限元离散求解,导出误差估计.最后,用数值例子说明算法的可行性和有效性.     

A posterior error analysis for the nonconforming discretization of Stokes eigenvalue problem

In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.     

Finite element error analysis of a variational multiscale method for the Navier-Stokes equations

The paper presents finite element error estimates of a variational multiscale method (VMS) for the incompressible Navier–Stokes equations. The constants in these estimates do not depend on the Reynolds number but on a reduced Reynolds number or on the mesh size of a coarse mesh. This work is partially supported by NSF grants DMS9972622, DMS20207627 and INT9814115.