Convergence analysis of sectional methods for solving breakage population balance equations-I: the fixed pivot technique

In this work we study the convergence of the fixed pivot techniques (Kumar and Ramkrishna Chem. Eng. Sci. 51, 1311–1332, 1996) for breakage problems. In particular, the convergence is investigated on four different types of uniform and non-uniform meshes. It is shown that the fixed pivot technique is second order convergent on a uniform and non-uniform smooth meshes. Furthermore, it gives first order convergence on a locally uniform mesh. Finally the analysis shows that the method does not converge on a non-uniform random mesh. The mathematical results of convergence analysis are also validated numerically.     

Convergence analysis of sectional methods for solving breakage population balance equations-II: the cell average technique

This work presents the convergence of the cell average technique (Kumar et al. in Powder Technol 179:205–228, 2007) for solving breakage population balance equation. Similarly to our paper Kumar and Warnecke (Numerische Math, 2008) of this series, we study convergence on four different types of meshes. A second order convergence is proved for uniform, locally uniform and non-uniform smooth meshes. Finally the scheme is analyzed on random mesh and it is found that the scheme is only first order accurate. Nevertheless we obtain for locally uniform as well as for random mesh one order higher accuracy than the fixed pivot technique discussed by the authors in the first paper. All mathematical observations of convergence analysis are also validated numerically and numerical results are compared with the results of the first part.     

High order finite difference methods on non-uniform meshes for space fractional operators

In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform meshes. The nonlocal property of space fractional operator makes it difficult to design the finite difference scheme on non-uniform meshes. This paper provides a basic strategy to derive the first and high order discretization schemes on non-uniform meshes for fractional operators. And the obtained first and second schemes on non-uniform meshes are used to solve space fractional diffusion equations. The error estimates and stability analysis are detailedly performed; and extensive numerical experiments confirm the theoretical analysis or verify the convergence orders.     

A finite difference scheme for solving the undamped Timoshenko beam equations with both ends free

In this paper, a boundary feedback system of a class of non-uniform undamped Timoshenko beam with both ends free is considered. A linearized three-level difference scheme for the Timoshenko beam equations is derived by the method of reduction of order on uniform meshes. The unique solvability, unconditional stability and convergence of the difference scheme are proved by the discrete energy method. The convergence order in maximum norm is of order two in both space and time. The validity of this theoretical analysis is verified experimentally.     

Convergence analysis of finite volume scheme for nonlinear aggregation population balance equation

In this work, we introduce the convergence analysis of the recently developed finite volume scheme to solve a pure aggregation population balance equation that is of substantial interest in many areas such as chemical engineering, aerosol physics, astrophysics, polymer science, pharmaceutical sciences, and mathematical biology. The notion of the finite volume scheme is to conserve total mass of the particles in the system by introducing weight in the formulation. The consistency of the finite volume scheme is also analyzed thoroughly as it is an influential factor. The convergence study of the numerical scheme shows second order convergence on uniform, nonuniform smooth (geometric) as well as on locally uniform meshes independent of the aggregation kernel. Moreover, the first‐order convergence is shown when the finite volume scheme is implemented on oscillatory and random meshes. In order to check the accuracy, the numerical experimental order of convergence is also computed for the physically relevant as well as analytically tractable kernels and validated against its analytical results.     

SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS

In this paper,we theoretically and numerically verify that the discontinuous Galerkin(DG)methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the L2-norm for even degree polynomial approximations.On uniform meshes,the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions,improving previous results.The theoretical findings are found to be sharp and consistent with numerical results.     

On the uniform convergence of a finite difference scheme for time dependent singularly perturbed reaction-diffusion problems

In this work we are interested in the numerical approximation of 1D parabolic singularly perturbed problems of reaction-diffusion type. To approximate the multiscale solution of this problem we use a numerical scheme combining the classical backward Euler method and central differencing. The scheme is defined on some special meshes which are the tensor product of a uniform mesh in time and a special mesh in space, condensing the mesh points in the boundary layer regions. In this paper three different meshes of Shishkin, Bahkvalov and Vulanovic type are used, proving the uniform convergence with respect to the diffusion parameter. The analysis of the uniform convergence is based on a new study of the asymptotic behavior of the solution of the semidiscrete problems, which are obtained after the time discretization by the Euler method. Some numerical results are showed corroborating in practice the theoretical results on the uniform convergence and the order of the method.     

Solving the order reduction phenomenon in variable step size quasi-consistent Nordsieck methods

The phenomenon is studied of reducing the order of convergence by one in some classes of variable step size Nordsieck formulas as applied to the solution of the initial value problem for a first-order ordinary differential equation. This phenomenon is caused by the fact that the convergence of fixed step size Nordsieck methods requires weaker quasi-consistency than classical Runge-Kutta formulas, which require consistency up to a certain order. In other words, quasi-consistent Nordsieck methods on fixed step size meshes have a higher order of convergence than on variable step size ones. This fact creates certain difficulties in the automatic error control of these methods. It is shown how quasi-consistent methods can be modified so that the high order of convergence is preserved on variable step size meshes. The regular techniques proposed can be applied to any quasi-consistent Nordsieck methods. Specifically, it is shown how this technique performs for Nordsieck methods based on the multistep Adams-Moulton formulas, which are the most popular quasi-consistent methods. The theoretical conclusions of this paper are confirmed by the numerical results obtained for a test problem with a known solution.     

The hp-MITC finite element method for the Reissner–Mindlin plate problem

The popular MITC finite elements used for the approximation of the Reissner–Mindlin plate are extended to the case where elements of non-uniform degree p distribution are used on locally refined meshes. Such an extension is of particular interest to the hp-version and hp-adaptive finite element methods. A priori error bounds are provided showing that the method is locking-free. The analysis is based on new approximation theoretic results for non-uniform Brezzi–Douglas–Fortin–Marini spaces, and extends the results obtained in the case of uniform order approximation on globally quasi-uniform meshes presented by Stenberg and Suri (SIAM J. Numer. Anal. 34 (1997) 544). Numerical examples illustrating the theoretical results and comparing the performance with alternative standard Galerkin approaches are presented for two new benchmark problems with known analytic solution, including the case where the shear stress exhibits a boundary layer. The new method is observed to be locking-free and able to provide exponential rates of convergence even in the presence of boundary layers.     

Convergence of the fixed pivot technique for continuous Smoluchowski coagulation equation

In this work, the fixed pivot technique (FPT) [2] is analyzed for nonlinear continuous Smoluchowski coagulation equation on four different types of grids. More importantly, the FPT gives the accuracy of second order for uniform and geometric grids while it reduces the order of accuracy by one on a locally uniform grid. At the end, the scheme is unfortunately zero order accurate on random grids. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Non-uniform Riemann Approach to Anticipating Stochastic Integrals

Abstract

In this note, we use the Riemann approach to give alternative definition to the anticipating stochastic integral. However non-uniform meshes instead of uniform meshes are used in the Riemann sums.     

A higher order difference method for singularly perturbed parabolic partial differential equations

This paper studies a higher order numerical method for the singularly perturbed parabolic convection-diffusion problems where the diffusion term is multiplied by a small perturbation parameter. In general, the solutions of these type of problems have a boundary layer. Here, we generate a spatial adaptive mesh based on the equidistribution of a positive monitor function. Implicit Euler method is used to discretize the time variable and an upwind scheme is considered in space direction. A higher order convergent solution with respect to space and time is obtained using the postprocessing based extrapolation approach. It is observed that the convergence is independent of perturbation parameter. This technique enhances the order of accuracy from first order uniform convergence to second order uniform convergence in space as well as in time. Comparative study with the existed meshes show the highly effective behavior of the present method.     

导数小片插值恢复技术与超收敛性

１．引言 有限元超收敛的研究自七十年代起至今方兴未艾．现有的研究工作基本遵循两种途径：一是找出有限元插值逼近的超收敛点,然后再利用插值弱估计等手段导出有限元解本身所具有的超收敛性质［１,２］;二是利用各种后处理技术,如平均技术,投影技术,外插技术和插值有限元技术等[3,6],来导出经过后处理的有限元解的超收敛性．近年来一种新的超收敛后处理技术,即所谓的“Ｚ-Ｚ导数小片恢复技术”,得到众多的研究［７-１１］,并被 Ｂａｂｕｓｋａ等人认为是用于渐进准确的后验误差估计效果最好的技术之一［１２］．这种技术是利用…     

Convergence analysis of a covolume scheme for Maxwell's equations in three dimensions

This paper contains error estimates for covolume discretizations of Maxwell's equations in three space dimensions. Several estimates are proved. First, an estimate for a semi-discrete scheme is given. Second, the estimate is extended to cover the classical interlaced time marching technique. Third, some of our unstructured mesh results are specialized to rectangular meshes, both uniform and nonuniform. By means of some additional analysis it is shown that the spatial convergence rate is one order higher than for the unstructured case.

     

奇异摄动问题内罚间断有限方法的最优阶一致收敛性分析

本文在Bakhvalov-Shishkin网格上分析了采用高次元的内罚间断有限元方法求解一维对流扩散型奇异摄动问题的最优阶一致收敛性. 取k(k≥1)次分片多项式和网格剖分单元数为N时,在能量范数度量下, Bakhvalov-Shishkin网格上可获得O(N^-k)的一致误差估计. 在数值算例部分对理论分析结果进行了验证.     

The Riemann approach to stochastic integration using non-uniform meshes

In this note we shall prove that the stochastic integral with respect to a semimartingale can be defined by Riemann's approach. However in this approach we use non-uniform meshes instead of the usual uniform meshes.     

The Derivative Patch Interpolating Recovery Technique for Finite Element Approximations

A derivative patch interpolating recovery technique is analyzed for the finite element approximation to the second order elliptic boundary value problems in two dimensional case.It is shown that the convergence rate of the recovered gradient admits superc onvergence on the recovered subdomain, and is two order higher than the optimal global convergence rate (ultracovergence) at an internal node point when even order finite element spaces and local uniform meshes are used.     

Gradient recovery type a posteriori error estimate for finite element approximation on non-uniform meshes

In this paper, we derive gradient recovery type a posteriori error estimate for the finite element approximation of elliptic equations. We show that a posteriori error estimate provide both upper and lower bounds for the discretization error on the non-uniform meshes. Moreover, it is proved that a posteriori error estimate is also asymptotically exact on the uniform meshes if the solution is smooth enough. The numerical results demonstrating the theoretical results are also presented in this paper.     

MULTIGRID METHODS FOR MORLEY ELEMENT ON NONNESTED MESHES

1.IntroductionWeconsidersomemultigridalgorithmsforthebiharmonicequationdiscretizedbyMoneyelementonnonnestedmeshes.TOdefineamultigridalgorithm,certainintergridtransferoperatorhastobeconstructed.Throughtakingtheaveragesofthenodalvariables,weconstructanintergridtransferoperatorforMoneyelementonnonnestedmeshesthatsatisfiesacertainstableapproximationpropertywhichplaysakeyroleinmultigridmethodsfornonconformingplateelementsonnonnestedmeshes.Theso--calledregularity-approximaticnassurnptionisestablis…     

A C0 interior penalty method for a singularly‐perturbed fourth‐order elliptic problem on a layer‐adapted mesh

We analyze the convergence of a continuous interior penalty (CIP) method for a singularly perturbed fourth‐order elliptic problem on a layer‐adapted mesh. On this anisotropic mesh, we prove under reasonable assumptions uniform convergence of almost order k ? 1 for finite elements of degree k ≥ 2. This result is of better order than the known robust result on standard meshes. A by‐product of our analysis is an analytic lower bound for the penalty of the symmetric CIP method. Finally, our convergence result is verified numerically. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 838–861, 2014