Determining surface temperature and heat flux by a wavelet dual least squares method

We apply a wavelet dual least squares method to a general sideways parabolic equation for determining surface temperature and surface heat flux. Connecting Meyer wavelet bases with a special project method dual least squares method, we can obtain a regularized solution. Meanwhile, order optimal error estimates between the approximate solution and exact solution are proved.     

A discontinuous Galerkin method for the Rosenau equation

A priori error estimates for the Rosenau equation, which is a K-dV like Rosenau equation modelled to describe the dynamics of dense discrete systems, have been studied by one of the authors. But since a priori error bounds contain the unknown solution and its derivatives, it is not effective to control error bounds with only a given step size. Thus we need to estimate a posteriori errors in order to control accuracy of approximate solutions using variable step sizes. A posteriori error estimates of the Rosenau equation are obtained by a discontinuous Galerkin method and the stability analysis is discussed for the dual problem. Numerical results on a posteriori error and wave propagation are given, which are obtained by using various spatial and temporal meshes controlled automatically by a posteriori error.     

A Dual Coupled Method for Boundary Value Problems of PDE with Coefficients of Small Period

1.IntroductionThemechanicalperformanceanalysisofthestructuresmadeofwovncompositematerialandperiodicallyperforatedmaterialisoftenencounteredinthemodernengi-neeringanalysis.Sincethiskindofcompositematerialhasperiodicallybasicconfigu-rations,thestaticanalysisofthestructuresmadefromthiscompositematerialleadstotheboundaryvalueproblemofellipticPDEwithperiodiccoefficients,forexample,theequllibriumproblemofwovenmembraneundertraverseloadingscanbeexpressedintheboundaryvalueproblemoftwodimensiontwoorder…     

A superconvergent fitted finite volume method for Black–Scholes equations governing European and American option valuation

We develop a superconvergent fitted finite volume method for a degenerate nonlinear penalized Black–Scholes equation arising in the valuation of European and American options, based on the fitting idea in Wang [IMA J Numer Anal 24 (2004), 699–720]. Unlike conventional finite volume methods in which the dual mesh points are naively chosen to be the midpoints of the subintervals of the primal mesh, we construct the dual mesh judiciously using an error representation for the flux interpolation so that both the approximate flux and solution have the second‐order accuracy at the mesh points without any increase in computational costs. As the equation is degenerate, we also show that it is essential to refine the meshes locally near the degenerate point in order to maintain the second‐order accuracy. Numerical results for both European and American options with constant and nonconstant coefficients will be presented to demonstrate the superconvergence of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1190–1208, 2015     

一维3×3非线性系统脉冲波的干扰

利用非线性几何光学的方法,讨论一维情形下3×3半线性偏微分方程具有两个初始脉冲波的干扰问题.证明方程组近似解的存在性,得到脉冲波的干扰出现在一阶项中.通过对近似解进行误差分析,得到精确解的一个好的近似.     

ITERATIVE METHODS FOR THE BOUNDARY VALUE PROBLEM OF A THIRD ORDER DIFFERENCE EQUATION

ＩＴＥＲＡＴＩＶＥＭＥＴＨＯＤＳＦＯＲＴＨＥＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＯＦＡＴＨＩＲＤＯＲＤＥＲＤＩＦＦＥＲＥＮＣＥＥＱＵＡＴＩＯＮＷａｎｇＰｅｉｇｕａｎｇ（王培光）（ＨｅｂｅｉＵｎｉｖｅｒｓｉｔｙ，河北大学，邮编：０７１００２）＆Ｌｕ...     

Further Exact and Approximate Considerations of the Barrier Problem

When the second order differential equation governing transmissionof waves through a potential barrier is solved approximately,two approximate solutions arise within the barrier, one exponentiallylarge and one exponentially small. When a linear combinationof these solutions is considered, the error involved in theexponentially large solution is much larger than the actualsmaller solution, leading to conceptual difficulties as to howthis small solution is to be interpreted within the linear combination.Here, a new approach is adopted, whereby linear combinationsof approximate solutions are avoided. The reflection coefficientof the barrier is derived and the series expansion of its modulusis obtained before approximations are introduced. The analysisis so arranged that ratios rather than linear combinations enterthis modulus, and error analysis then shows exactly why theerror consists of certain unexpected exponentially small termsrather than expected terms of larger order of magnitude.     

Bernstein polynomial basis for numerical solution of boundary value problems

The purpose of this paper is to propose a computational method for the approximate solution of linear and nonlinear two-point boundary value problems. In order to approximate the solution, the expansions in terms of the Bernstein polynomial basis have been used. The properties of the Bernstein polynomial basis and the corresponding operational matrices of integration and product are utilized to reduce the given boundary value problem to a system of algebraic equations for the unknown expansion coefficients of the solution. On this approach, the problem can be solved as a system of algebraic equations. By considering a special case of the problem, an error analysis is given for the approximate solution obtained by the present method. At last, five examples are examined in order to illustrate the efficiency of the proposed method.     

Stokes问题Q_2-P_1混合元外推方法

考虑Stokes问题的有限元解与精确解插值的Q2-P1混合元的渐近误差展开和分裂外推.首先利用积分恒等式技巧确定了微分方程精确解与有限元插值之间积分式的主项,其次再借助插值后处理和分裂外推技术,得到了比通常的误差估计提高两阶的收敛速度.     

Finite element solution of nonlinear diffusion problems

In this paper we describe the Rothe-finite element numerical scheme to find an approximate solution of a nonlinear diffusion problem modeled as a parabolic partial differential equation of even order. This scheme is based on the Rothe’s approximation in time and on the finite element method (FEM) approximation in the spatial discretization. A proof of convergence of the approximate solution is given and error estimates are shown.     

An iterative algorithm for large size least-squares constrained regularization problems

In this paper we propose an iterative algorithm to solve large size linear inverse ill posed problems. The regularization problem is formulated as a constrained optimization problem. The dual Lagrangian problem is iteratively solved to compute an approximate solution. Before starting the iterations, the algorithm computes the necessary smoothing parameters and the error tolerances from the data.The numerical experiments performed on test problems show that the algorithm gives good results both in terms of precision and computational efficiency.     

A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.     

Identification of an inverse source problem for time‐fractional diffusion equation with random noise

In this paper, we consider an inverse source problem for a time fractional diffusion equation. In general, this problem is ill posed, therefore we shall construct a regularized solution using the filter regularization method in the random noise case. We will provide appropriate conditions to guarantee the convergence of the approximate solution to the exact solution. Then, we provide examples of filters in order to obtain error estimates for their approximate solutions. Finally, we present a numerical example to show efficiency of the method.     

二阶椭圆问题新混合元模型的超收敛分析及外推

对二阶椭圆问题通过＂增补＂办法导出一个新的混合模型.在各向异性网格下,利用积分恒等式技巧得到了真解与ECHL元近似解的超逼近性质.同时基于插值后处理技术导出了整体超收敛.进一步,通过渐进误差展开和分裂外推,得到了比通常的误差估计更高一阶的收敛速度.     

Goal-oriented r-adaptivity based on the evaluation of the physical and material residual

This contribution is concerned with goal–oriented r-adaptivity based on energy minimization principles for the primal and the dual problem. We obtain a material residual of the primal and of the dual problem, which are indicators for non–optimal finite element meshes. For goal–oriented r-adaptivity we have to optimize the mesh with respect to the dual solution, because the error of a local quantity of interest depends on the error in the corresponding dual solution. We use the material residual of the primal and dual problem in order to obtain a procedure for mesh optimization with respect to a local quantity of interest. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time integration error estimation for continuous Galerkin schemes

The purpose of this contribution is the time integration error estimation for continuous Galerkin schemes applied to the linear semi-discrete equation of motion. A special focus is on the effort for the error estimation for large finite element models. Error estimators for the global time integration error as well as for the local error in the last time interval are presented. The Galerkin formulation in time allows the application of the well-known duality based error estimation techniques for the estimation of the time integration error. The main effort of these error estimators is the computation of the dual solution. In order to diminish the computational effort for solving the dual problem the error estimation is carried out in a reduced modal basis. The relevant modes which have to remain in the basis can be determined via the initial conditions of the dual problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Modified nodal cubic spline collocation for elliptic equations

We consider the modified nodal cubic spline collocation method for a general, variable coefficient, second order partial differential equation in the unit square with the solution subject to the homogeneous Dirichlet boundary conditions. The bicubic spline approximate solution satisfies both the Dirichlet boundary conditions and a perturbed partial differential equation at the nodes of a uniform partition of the square. We prove existence and uniqueness of the approximate solution and derive an optimal fourth order maximum norm error bound. The resulting linear system is solved efficiently by a preconditioned iterative method. Numerical results confirm the expected convergence rates. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Constructing regularization methods in the spaces of differentiable functions

The method of finding approximations to a solution and its derivatives is constructed for a certain class of integral Volterra equations of the first kind. Matching conditions for the regularization parameter and the error in the initial data are presented. Sharp (in terms of order) error estimates are obtained for approximate solutions on certain compact classes.