The superconvergence of the composite midpoint rule for the finite-part integral

AbstractThe composite midpoint rule is probably the simplest one among the Newton–Cotes rules for Riemann integral. However, this rule is divergent in general for Hadamard finite-part integral. In this paper, we turn this rule to a useful one and, apply it to evaluate Hadamard finite-part integral as well as to solve the relevant integral equation. The key point is based on the investigation of its pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate of the midpoint rule is higher than what is globally possible. We show that the superconvergence rate of the composite midpoint rule occurs at the midpoint of each subinterval and obtain the corresponding superconvergence error estimate. By applying the midpoint rule to approximate the finite-part integral and by choosing the superconvergence points as the collocation points, we obtain a collocation scheme for solving the finite-part integral equation. More interesting is that the inverse of the coefficient matrix of the resulting linear system has an explicit expression, by which an optimal error estimate is established. Some numerical examples are provided to validate the theoretical analysis

The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernel cot((x-s)/2) is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis

Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

In this paper, we study the convergence of the finite difference discretization of a second order elliptic equation with variable coefficients subject to general boundary conditions. We prove that the scheme exhibits the phenomenon of supraconvergence on nonuniform grids, i.e., although the truncation error is in general of the first order alone, one has second order convergence. All error estimates are strictly local. Another result of the paper is a close relationship between finite difference scheme and linear finite element methods combined with a special kind of quadrature. As a consequence, the results of the paper can be viewed as the introduction of a fully discrete finite element method for which the gradient is superclose. A numerical example is given

Numerical methods to compute hypersingular integral in boundary element methods

超奇异积分的近似计算是边界元方法,特别是自然边界元理论中必须面对的难题之一.经典的数值方法,如gAuSS求积公式和nEWTOn-COTES积分公式等数值方法,都不能直接用于超奇异积分的近似计算.本文将介绍超奇异积分基于不同定义的gAuSS积分公式、S型变换公式、nEWTOn-COTES积分公式和外推法近似计算超奇异积分的思路,重点阐述nEWTOn-COTES积分公式和基于有限部分积分定义的外推法近似计算超奇异积分的主要结论.The computation of hypersingular integral is one of the important subjects in boundary element methods especially in natural boundary element methods.Classical numerical methods such as Gauss methods,Newton-Cotes methods cannot be used to approximate the hypersingular integral directly.In this paper, we introduce the numerical methods such as Gauss methods, Newton-Cotes methods, S transformation methods and extrapolation methods which are based on different definitions; then we mainly present the results of Newton-Cotes methods and extrapolation methods which are used to compute the hypersingular integral.国家自然科学基金(批准号:11471195;11101247;11201209和91330106); 中国博士后科学基金(批准号:2013M540541)资助项

Multiblock modeling of flow in porous media and applications

We investigate modeling flow in porous media in multiblock domain. Mixed finite element methods are used for subdomain discretizations. Physically meaningful boundary conditions are imposed on the non-matching interfaces via mortar finite element spaces. We investigate the pollution effect of nonmatching grids error on the numerical solution away from interfaces. We prove that most of the error in the velocity occurs along the interfaces, and that high accuracy is preserved in the interior of the subdomains. In case of discontinuous coefficients, the pollution from the singularity affects the accuracy in the whole domain. We investigate the upscaling error resulting when fine resolution data is approximated on a very coarse scale. Extending work of Wheeler and Yotov, we incorporate this upscaling error in an a posteriori error estimator for the pressure, velocity and mortar pressure. We employ a non-overlapping domain decomposition method reducing the global system to one that is solved iteratively via a preconditioned conjugate gradient method. This approach is suitable for parallel implementation. The balancing domain decomposition method for mixed finite elements following Cowsar, Mandel, and Wheeler is extended to the case of mortar mixed finite elements on non-matching multiblock grids. The algorithm involves solution of a mortar interface problem with one local Dirichlet solve and one local Neumann solve on each iteration. A coarse solve is used to guarantee consistency and to provide global exchange of information. Quasi-optimal condition number bounds independent of the jump in coefficients are derived. We finally consider multiscale mortar mixed finite element discretizations for single and two phase flows. We show optimal convergence and some superconvergence in the fine scale for the solution and its flux. We also derive efficient and reliable a posteriori error estimators suitable for adaptive mesh refinement. We have incorporated the above methods into a parallel multiblock simulator on unstructured prismatic meshes employing a non-overlapping domain decomposition algorithm and mortar spaces. Numerical experiments are presented confirming all theoretical results