An optimal‐order error estimate for MMOC and MMOCAA schemes for multidimensional advection‐reaction equations

In this article, we analyze the modified method of characteristics (MMOC) and an improved version of the MMOC, named the modified method of characteristics with adjusted advection (MMOCAA), for multidimensional advection‐reaction transport equations in a uniform manner. We derive an optimal‐order error estimate for these schemes. Numerical results are presented to verify the theoretical estimates. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 69–84, 2002     

A uniform estimate for the MMOC for two‐dimensional advection‐diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for the modified method of characteristics (MMOC) and the modified method of characteristics with adjusted advection (MMOCAA) for two‐dimensional time‐dependent advection‐diffusion equations, in the sense that the generic constants in the estimates depend on certain Sobolev norms of the true solution but not on the scaling diffusion parameter ε. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

KdV方程的特征线混合间断有限元方法

考虑KdV方程的两种特征线性混合间断有限元方法,一种方法建立在标准特征线修正方法的基础上,另一种方法是带有对流项修正的特征线修正方法.利用具有实际物理意义的特征线追踪技巧处理时间导数项和对流项,采用混合间断有限元方法处理三阶导数项,分别证明了两种方法有限元解的唯一性、稳定性和误差估计.     

Error estimates for finite volume element methods for general second‐order elliptic problems

We treat the finite volume element method (FVE) for solving general second order elliptic problems as a perturbation of the linear finite element method (FEM), and obtain the optimal H¹ error estimate, H¹ superconvergence and L^p (1 < p ≤ ∞) error estimates between the solution of the FVE and that of the FEM. In particular, the superconvergence result does not require any extra assumptions on the mesh except quasi‐uniform. Thus the error estimates of the FVE can be derived by the standard error estimates of the FEM. Moreover we consider the effects of numerical integration and prove that the use of barycenter quadrature rule does not decrease the convergence orders of the FVE. The results of this article reveal that the FVE is in close relationship with the FEM. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 693–708, 2003.     

A Symmetric Characteristic Finite Volume Element Scheme for Nonlinear Convection-Diffusion Problems

In this paper, we implement alternating direction strategy and construct a symmetric FVE scheme for nonlinear convection-diffusion problems. Comparing to general FVE methods, our method has two advantages. First, the coefficient matrices of the discrete schemes will be symmetric even for nonlinear problems. Second, since the solution of the algebraic equations at each time step can be inverted into the solution of several one-dimensional problems, the amount of computation work is smaller. We prove the optimal H1-norm error estimates of order O（△t2 ＋ h） and present some numerical examples at the end of the paper.     

Analysis and application of finite volume element methods to a class of partial differential equations

The finite volume element (FVE) methods for a class of partial differential equations are discussed and analyzed in this paper. The new initial values are introduced in the finite volume element schemes, and we obtain optimal error estimates in Lp and W1,p (2?p?∞) as well as some superconvergence estimates in W1,p (2?p?∞). The main results in this paper perfect the theory of the finite volume element methods.     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

A symmetric characteristic FVE method with second order accuracy for nonlinear convection diffusion problems

The finite volume element (FVE) methods used currently are essentially low order and unsymmetric. In this paper, by biquadratic elements and multistep methods, we construct a second order FVE scheme for nonlinear convection diffusion problem on nonuniform rectangular meshes. To overcome the numerical oscillation, we discretize the problem along its characteristic direction. The choice of alternating direction strategy is critical in this paper, which guarantees the high efficiency and symmetry of the discrete scheme. Optimal order error estimates in H¹$H^1$-norm are derived and a numerical example is given at the end to confirm the usefulness of the method.     

Error estimates of H 1-Galerkin mixed finite element method for Schrödinger equation

An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.     

Error estimates of the finite element method for the exterior Helmholtz problem with a modified DtN boundary condition

A priori error estimates in the H¹- and L²-norms are established for the finite element method applied to the exterior Helmholtz problem, with modified Dirichlet-to-Neumann (MDtN) boundary condition. The error estimates include the effect of truncation of the MDtN boundary condition as well as that of discretization of the finite element method. The error estimate in the L²-norm is sharper than that obtained by the author [D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math. 200 (1) (2007) 21-31] for the truncated DtN boundary condition.     

Optimal l~∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions

Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction.The methods which we considered include two conservative type schemes and two non-conservative type schemes.Discrete conservation laws and solvability of the schemes are analyzed.For the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h~2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size h.Besides the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order reduction.All the methods and results here are also valid and can be easily extended to the three-dimensional case.Finally,numerical results are reported to confirm our theoretical error estimates for the numerical methods.     

Modified form of binary and ternary 3-point subdivision schemes

Binary 3-point scheme, developed by Hormann and Sabin [Hormann, K. and Sabin, Malcolm A., 2008, A family of subdivision schemes with cubic precision, Computer Aided Geometric Design, 25, 41-52], has been modified by introducing a tension parameter which generates a family of C¹ limiting curves for certain range of tension parameter. Ternary 3-point scheme, introduced by Siddiqi and Rehan [Siddiqi, Shahid S. and Rehan, K., 2009, A ternary three point scheme for curve designing, International Journal of Computer Mathematics, In Press, DOI: 10.1080/00207160802428220], has also been modified by introducing a tension parameter which generates family of C¹ and C² limiting curves for certain range of tension parameter. Laurent polynomial method is used to investigate the continuity of the subdivision schemes. The performance of modified schemes has been demonstrated by considering different examples along with its comparison with the established subdivision schemes.     

非线性对流扩散方程的UNO-MMOCAA差分方法

本文把MMOCAA差分方法与UNO插值相结合，提出了求解对流占优扩散问题的UN0—MMOCAA差分方法，它避免了基于高次(≥2)Lagrange插值的MMOCAA差分方法在方程解的陡峭前沿附近产生的振荡．本文通过引入辅助插值算于等方法，给出了非线性UNO—MMOCAA差分格式的误差分析．数值例子表明新格式无振荡。     

Unconditional and optimal <Emphasis Type="Bold">H</Emphasis><Superscript>2</Superscript>-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions

The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear Klein-Gordon-Schrödinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H ²-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h ² + τ ²) with mesh-size h and time step τ in the discrete H ²-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.     

Optimal error estimates of mixed finite element methods for a fourth-order nonlinear elliptic problem

A mixed finite element method is developed for a nonlinear fourth-order elliptic problem. Optimal L² error estimates are proved by using a special interpolation operator on the standard tensor-product finite elements of order k?1. Then two iterative schemes are presented and proved to keep the same optimal error estimates. Three numerical examples are provided to support the theoretical analysis.     

A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems

A proper orthogonal decomposition (POD) method is applied to a usual finite volume element (FVE) formulation for parabolic equations such that it is reduced to a POD FVE formulation with lower dimensions and high enough accuracy. The error estimates between the reduced POD FVE solution and the usual FVE solution are analyzed. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced POD FVE formulation based on POD method is both feasible and highly efficient.     

非饱和土壤水流问题基于POD 方法的降阶有限体积元格式及外推算法实现

利用特征投影分解(proper orthogonal decomposition, 简记为POD) 方法对非饱和土壤水流问题的经典有限体积元格式做降阶处理, 建立一种具有足够高精度维数较低的降阶有限体积元格式, 并给出这种降阶有限体积元解的误差估计和外推算法的实现, 最后用数值例子说明数值结果与理论结果是相吻合的. 进一步表明了基于POD 方法的降阶有限体积元格式对求解非饱和土壤水流问题数值解是可靠和有效的.     

A new mixed scheme based on variation of constants for Sobolev equation with nonlinear convection term

A new mixed scheme which combines the variation of constants and the H ¹-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear convection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.     

A New Reduced-Order fve Algorithm Based on POD Method for Viscoelastic Equations

A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced-order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illustrate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.