Numerical analysis of a non‐singular boundary integral method: Part II: The general case

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we have presented in a previous paper a method which consists in inverting, on a finite element space, a non‐singular integral operator for circular domains. This operator was described as a geometrical perturbation of the Steklov operator, and we have precisely defined the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme in which there are non‐singular integrals. We have also presented another point of view under which the method can be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single‐layer potential. In the present paper, we extend the results given in the previous paper to more general cases for which the Laplace problem is set on any ??^∞ domains. We prove that the properties of stability and convergence remain valid. Copyright © 2002 John Wiley & Sons, Ltd.     

计算高维带弱奇异核发展型方程的交替方向隐式欧拉方法

本文主要研究高维带弱奇异核的发展型方程的交替方向隐式(ADI)差分方法.向后欧拉(Euler)方法联立一阶卷积求积公式处理时间方向的离散,有限差分方法处理空间方向的离散,并进一步构造了ADI全离散差分格式.然后将二维问题延伸到三维问题,构造三维空间问题的ADI差分格式.基于离散能量法,详细证明了全离散格式的稳定性和误差分析.随后给出了2个数值算例,数值结果进一步验证了时间方向的收敛阶为一阶,空间方向的收敛阶为二阶,和理论分析结果一致.     

Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations

The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.     

A novel finite volume method for the Riesz space distributed-order advection–diffusion equation

In this paper, we investigate the finite volume method (FVM) for a distributed-order space-fractional advection–diffusion (AD) equation. The mid-point quadrature rule is used to approximate the distributed-order equation by a multi-term fractional model. Next, the transformed multi-term fractional equation is solved by discretizing in space by the finite volume method and in time using the Crank–Nicolson scheme. We use a novel technique to deal with the convection term, by which the Riesz fractional derivative of order 0 < γ < 1 is transformed into a fractional integral form. An important contribution of our work is the use of nodal basis function to derive the discrete form of our model. The unique solvability of the scheme is also discussed and we prove that the Crank–Nicolson scheme is unconditionally stable and convergent with second-order accuracy. Finally, we give some examples to show the effectiveness of the numerical method.     

NON C~0 NONCONFORMING ELEMENTS FOR ELLIPTIC FOURTH ORDER SINGULAR PERTURBATION PROBLEM

In this paper we give a convergence theorem for non C^0 nonconforming finite element to solve the elliptic fourth order singular perturbation problem. Two such kind of elements, a nine parameter triangular element and a twelve parameter rectangular element both with double set parameters, are presented. The convergence and numerical results of the two elements are given.     

Estimating quadrature errors for an efficient method for quasi-singular boundary integral

The numerical resolution of the boundary integral equations applied to the differential equations of Laplace, Helmholtz and Maxwell requires the handling of quasi-singular integrals with different order of singularity. The numerical approximation of the integral equations of different kinds is made by boundary finite elements. In this paper, we present a complete survey for estimating quadrature errors for the numerical techniques proposed by Huang and Cruse [Q. Huang, T.A. Cruse, Some notes on singular integral techniques in boundary element analysis, Int. J. Numer. Methods Eng. 36 (15) (1993) 2643-2659], to calculate the quasi-singular integrals. To validate the accuracy and efficiency of these techniques and approve our study some numerical examples are presented and discussed.     

A parameter-uniform numerical method for a Sobolev problem with initial layer

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.      

ERROR ESTIMATES FOR SPARSE OPTIMAL CONTROL PROBLEMS BY PIECEWISE LINEAR FINITE ELEMENT APPROXIMATION

Optimization problems with L¹-control cost functional subject to an elliptic partial differential equation(PDE)are considered.However,different from the finite dimensiona l¹-regularization optimization,the resulting discretized L¹norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem.A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the L¹-norm.In this paper,a new discretized scheme for the L¹-norm is presented.Compared to the new discretized scheme for L¹-norm with the nodal quadrature formula,the advantages of our new discretized scheme can be demonstrated in terms of the order of approximation.Moreover,finite element error estimates results for the primal problem with the new discretized scheme for the L¹-norm are provided,which confirms that this approximation scheme will not change the order of error estimates.To solve the new discretized problem,a symmetric Gauss-Seidel based majorized accelerated block coordinate descent(sGS-mABCD)method is introduced to solve it via its dual.The proposed sGS-mABCD algorithm is illustrated at two numerical examples.Numerical results not only confirm the finite element error estimates,but also show that our proposed algorithm is efficient.     

A second order scheme for a time-dependent,singularly perturbed convection–diffusion equation

We consider a numerical scheme for a one-dimensional, time-dependent, singularly perturbed convection–diffusion problem. The problem is discretized in space by a standard finite element method on a Bakhvalov–Shishkin type mesh. The space error is measured in an L² norm. For the time integration, the implicit midpoint rule is used. The fully discrete scheme is shown to be convergent of order 2 in space and time, uniformly in the singular perturbation parameter.     

High-order dual-parametric finite element methods for cavitation computation in nonlinear elasticity

In this paper, we present the numerical analysis on high order dual parametric finite element methods for the cavitation computation problems in nonlinear elasticity, which leads to a meshing strategy assuring high efficiency on numerical approximations to cavity deformations. Furthermore, to cope with the high order approximation of the finite element methods, properly chosen weighted Gaussian type numerical quadrature is applied to the singular part of the elastic energy. Our numerical experiments show that the high order dual parametric finite element methods work well when coupled with properly designed weighted Gaussian type numerical quadratures for the singular part of the elastic energy, and the convergence rates of the numerical cavity solutions are shown to be significantly improved as expected.     

Error estimates and extrapolation for the numerical solution of Mellin convolution equations

In this paper we consider a quadrature method for the numericalsolution of a second-kind integral equation over the interval,where the integral operator is a compact perturbation of a Mellinconvolution operator. This quadrature method relies upon a singularitysubtraction and transformation technique. Stability and convergenceorder of the approximate solution are well known. We shall derivethe first term in the asymptotics of the error which shows that,in the interior of the interval, the approximate solution convergeswith higher order than over the whole interval. This implieshigher orders of convergence for the numerical calculation ofsmooth functionals to the exact solution. Moreover, the asymptoticsallows us to define a new approximate solution extrapolatedfrom the dilated solutions of the quadrature method over mesheswith different mesh sizes. This extrapolated solution is designedto improve the low convergence order caused by the non-smoothnessof the exact solution even when the transformation techniquecorresponds to slightly graded meshes. Finally, we discuss theapplication to the double-layer integral equation over the boundaryof polygonal domains and report numerical results.     

A Compact Difference Scheme for an Evolution Equation with a Weakly Singular Kernel

This paper is concerned with a compact difference scheme with the truncation error of order 3/2 for time and order 4 for space to an evolution equation with a weakly singular kernel. The integral term is treated by means of the second order convolution quadrature suggested by Lubich. The stability and convergence are proved by the energy method. A numerical experiment is reported to verify the theoretical predictions.     

On adaptive mesh for the initial boundary value singularly perturbed delay Sobolev problems

A uniform finite difference method on a B-mesh is applied to solve the initial-boundary value problem for singularly perturbed delay Sobolev equations. To solve the foresold problem, finite difference scheme on a special nonuniform mesh, whose solution converges point-wise independently of the singular perturbation parameter is constructed and analyzed. The present paper also aims at discussing the stability and convergence analysis of the method. An error analysis shows that the method is of second order convergent in the discrete maximum norm independent of the perturbation parameter. A numerical example and the simulation results show the effectiveness of our theoretical results.     

Fourth‐order compact scheme with local mesh refinement for option pricing in jump‐diffusion model

The value of a contingent claim under a jump‐diffusion process satisfies a partial integro‐differential equation. A fourth‐order compact finite difference scheme is applied to discretize the spatial variable of this equation. It is discretized in time by an implicit‐explicit method. Meanwhile, a local mesh refinement strategy is used for handling the nonsmooth payoff condition. Moreover, the numerical quadrature method is exploited to evaluate the jump integral term. It guarantees a Toeplitz‐like structure of the integral operator such that a fast algorithm is feasible. Numerical results show that this approach gives fourth‐order accuracy in space. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

A quadrature finite element method for semilinear second‐order hyperbolic problems

In this work we propose and analyze a fully discrete modified Crank–Nicolson finite element (CNFE) method with quadrature for solving semilinear second‐order hyperbolic initial‐boundary value problems. We prove optimal‐order convergence in both time and space for the quadrature‐modified CNFE scheme that does not require nonlinear algebraic solvers. Finally, we demonstrate numerically the order of convergence of our scheme for some test problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS

This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations.The proposed method is based on the mixed finite element method in space and a finite difference scheme in time.The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail.Furthermore,We give the convergence analysis for both semidiscrete and flly discrete schemes and then prove that the numerical solution converges the exact one with order O(h2+k),where h and k:respectively denote the space step size and the time step size.Finally,numerical examples are presented to demonstrate the effectiveness of our numerical methods.     

THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS

1 Introduction Singular integral equations (SIEs) with Cauchy kernels Of the formoften arise in mathematical models of physical phenomena. Since closed-form solutions to SIEsare generally not available, much att.ntion has been focused on numerical methods of solution.In the past twenty years, various collocation methods for SIEs have been the topic of a greatmany of papers, most of which can be found in two surveys[213]. The early works in the fieldis to study tile numerical solutions for…     

SINGULAR INTEGRAL OPERATORS AND SINGULAR QUADRATURE OPERATORS ASSOCIATED WITH SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND AND THEIR APPLICATIONS

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Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWF). This set of special functions have been defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super‐exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.     

A Symmetric Galerkin Boundary Element Method for Linear Poroelasticity

In linear poroelasticity so far only collocation boundary element methods have been available. However, in some applications, e.g., when coupling with finite elements is desired, a symmetric formulation is preferable. Choosing a Galerkin approach which involves the second boundary integral equation, such a formulation is possible. Here, a previously presented integration by part technique for the regularization of the first boundary integral equation is extended to the second boundary integral equation as well. While the weakly singular representation of the double layer operator has been presented before, the emphasis lies here on the so called hyper-singular boundary integral operator. Due to the regularization, this operator can be evaluated numerically and, hence, be used within a numerical scheme for the first time. Different numerical studies will be presented to show the behavior of the established symmetric Galerkin boundary element method, also comparing it with collocation boundary element methods. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)