Theoretical Error Estimates Research Articles

A meshless local discrete collocation (MLDC) scheme for solving 2‐dimensional singular integral equations with logarithmic kernels

AbstractIn the current investigation, we study a numerical technique to solve 2‐dimensional singular integral equations of the second kind with logarithmic kernels. These integral equations are a general form of the problem of obtaining the cross‐sectional separation of current in an infinitely long thin filament of conduct bar which arises in electrical engineering. The moving least squares scheme estimates a function without mesh generation on the domain that includes a locally weighted least squares polynomial fitting. The discrete collocation method in addition to the shape functions of moving least squares established on scattered points is used to approximate the solution of integral equations. To compute logarithm‐like singular integrals appeared in the process of the scheme, we use a particular dual nonuniform composite Gauss‐Legendre integration rule on normal domains. The proposed scheme does not require any meshes, so it is meshless and does not depend to the domain form. 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.

Liquid water path retrieval using the lowest frequency channels of Fengyun-3C Microwave Radiation Imager (MWRI)

The Microwave Radiation Imager (MWRI) on board Chinese Fengyun-3 (FY-3) satellites provides measurements at 10.65, 18.7, 23.8, 36.5, and 89.0 GHz with both horizontal and vertical polarization channels. Brightness temperature measurements of those channels with their central frequencies higher than 19 GHz from satellite-based microwave imager radiometers had traditionally been used to retrieve cloud liquid water path (LWP) over ocean. The results show that the lowest frequency channels are the most appropriate for retrieving LWP when its values are large. Therefore, a modified LWP retrieval algorithm is developed for retrieving LWP of different magnitudes involving not only the high frequency channels but also the lowest frequency channels of FY-3 MWRI. The theoretical estimates of the LWP retrieval errors are between 0.11 and 0.06 mm for 10.65- and 18.7-GHz channels and between 0.02 and 0.04 mm for 36.5- and 89.0-GHz channels. It is also shown that the brightness temperature observations at 10.65 GHz can be utilized to better retrieve the LWP greater than 3 mm in the eyewall region of Super Typhoon Neoguri (2014). The spiral structure of clouds within and around Typhoon Neoguri can be well captured by combining the LWP retrievals from different frequency channels.

A meshless Galerkin scheme for the approximate solution of nonlinear logarithmic boundary integral equations utilizing radial basis functions

This paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations. These types of integral equations arise from boundary value problems of Laplace’s equations with nonlinear Robin boundary conditions. The discrete Galerkin method together with the (inverse) multiquadric radial basis functions established on scattered points is utilized to approximate the solution. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method. The proposed scheme uses a special accurate quadrature formula via the nonuniform Gauss–Legendre integration rule to compute logarithm-like singular integrals appeared in the scheme. Since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain, it is meshless and does not depend to the domain form. We also investigate the error analysis of the proposed method. Illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

Solving weakly singular integral equations utilizing the meshless local discrete collocation technique

The current work presents a computational scheme to solve weakly singular integral equations of the second kind. The discrete collocation method in addition to the moving least squares (MLS) technique established on scattered points is utilized to estimate the solution of integral equations. The MLS scheme approximates a function without mesh refinement over the domain that includes a locally weighted least squares polynomial fitting. The discrete collocation technique for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize an accurate quadrature formula based on the use of non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the approach. The proposed scheme does not require any meshes, so it can be called as the meshless local discrete collocation (MLDC) 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.

Solving a class of nonlinear boundary integral equations based on the meshless local discrete Galerkin (MLDG) method

The main purpose of this article is to investigate a computational scheme for solving a class of nonlinear boundary integral equations which occurs as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method approximates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least square polynomial fitting. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals appeared in the method. The numerical scheme developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule to estimate logarithm-like singular integrals. Since the proposed method is constructed on a set of scattered points, it does not require any background mesh and so we can call it as the meshless local discrete Galerkin (MLDG) method. The scheme is simple and effective to solve boundary integral equations and its algorithm can be easily implemented. We also obtain the error bound and the convergence rate of the presented method. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.

Error analysis of a least squares pseudo-derivative moving least squares method

Meshfree methods offer the potential to relieve the scientist from the time consuming grid generation process especially in cases where localized mesh refinement is desired. Moving least squares (MLS) methods are considered such a meshfree technique. The pseudo-derivative (PD) approach has been used in many papers to simplify the manipu- lations involved in MLS schemes. In this paper, we provide theoretical error estimates for a least squares implementation of an MLS/PD method with a stabilization mechanism. Some beginning computations suggest this stabilization leads to good matrix conditioning.

Improved complex variable moving least squares approximation for three-dimensional problems using boundary integral equations

The improved complex variable moving least squares approximation is an efficient method to generate meshless approximation functions. In the past, the approximation has been used only for 2D problems. In this paper, the approximation is developed to solve 3D problems. Theoretical error estimation of the approximation is given. Then, incorporating the approximation into boundary integral equations, a symmetric and boundary-only meshless method, the complex variable Galerkin boundary node method, is developed and analyzed theoretically for 3D potential, Helmholtz and Stokes problems. Numerical results demonstrate the accuracy and efficiency of the developed method.

Numerical Approximation of Space-Time Fractional Parabolic Equations

Abstract In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e. an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator E ⁢ ( t ) {E(t)} for the initial value problem can be written as a Dunford–Taylor integral involving the Mittag-Leffler function e α , 1 {e_{\alpha,1}} and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here α denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator W ⁢ ( t ) {W(t)} and the forcing function F ⁢ ( t ) {F(t)} . We develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford–Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of 1 h {\frac{1}{h}} . The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying a sinc quadrature technique to approximate the Dunford–Taylor integral of the discrete operator and is free of any time stepping. The sinc quadrature of step size k involves k - 2 {k^{-2}} nodes and results in an additional O ⁢ ( exp ⁡ ( - c k ) ) {O(\exp(-\frac{c}{k}))} error. To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo-midpoint quadrature. This involves the average of W h ⁢ ( s ) {W_{h}(s)} , (the semi-discrete approximation to W ⁢ ( s ) {W(s)} ) over the quadrature interval. This average can also be written as a Dunford–Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford–Taylor integrals for computing the averages. We show that for a refined grid in time with a mesh of O ⁢ ( 𝒩 ⁢ log ⁡ ( 𝒩 ) ) {O({\mathcal{N}}\log({\mathcal{N}}))} intervals, the error between the semi-discrete and fully discrete approximation is O ⁢ ( 𝒩 - 2 + log ⁡ ( 𝒩 ) ⁢ exp ⁡ ( - c k ) ) {O({\mathcal{N}}^{-2}+\log({\mathcal{N}})\exp(-\frac{c}{k}))} . We also report the results of numerical experiments that are in agreement with the theoretical error estimates.

A meshless complex variable Galerkin boundary node method for potential and Stokes problems

In this study, combining the boundary integral equations (BIEs) with the complex variable moving least squares (CVMLS) approximation, a symmetric and boundary-only meshless method, the complex variable Galerkin boundary node method (CVGBNM), is developed. Numerical applications and theoretical error estimates of the CVGBNM are derived for BIEs, potential problems and Stokes problems. Finally, numerical examples are given to demonstrate the efficacy of the method.

Numerical integration based on hyperfunction theory

In this paper, we propose an application of hyperfunction theory to numerical integration. Hyperfunction theory is a generalized version of function theory where functions with singularities such as poles, discontinuities and delta impulses are expressed in terms of complex analytic functions. This feature of hyperfunction theory allows us to construct a numerical integration method for analytic functions. Theoretical error estimates show that our method converges geometrically, and numerical examples show that our method is very efficient especially for integrals with strong end-point singularities. In addition, we present an automatic integration method based on our hyperfunction method.

A meshless method for the numerical solution of nonlinear weakly singular integral equations using radial basis functions

In this paper a simple and effective method is given for the numerical solution of nonlinear one- and two-dimensional Fredholm integral equations of the second kind with weakly singular kernels. The general framework of the new scheme is based on the collocation method together with radial basis functions (RBFs) constructed on scattered points in which all integrals are computed via quadrature formulae. In order to approximate the singular integrals appeared in the scheme, we introduce a special quadrature formula, since these integrals cannot be estimated by classical integration rules. The method does not require any cell structures, so it is meshless and consequently is independent of the geometry of the domain. We also present the error analysis of the proposed method and demonstrate that the convergence rate of the approach is arbitrarily high for infinitely smooth RBFs. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.

The numerical solution of two-dimensional logarithmic integral equations on normal domains using radial basis functions with polynomial precision

The main purpose of this article is to investigate the numerical solution of two-dimensional Fredholm integral equations of the second kind on normal domains, whose kernels have logarithmic singularity. Radial basis functions constructed on scattered points are utilized as a basis in the discrete collocation method to solve these types of integral equations. We encounter logarithm-like singular integrals in the process of setting up the presented scheme which cannot be computed by classical quadrature formulae. Therefore, a special numerical integration rule is required to approximate such integrals based on the use of dual non-uniform composite Gauss---Legendre quadratures on normal domains. Since the method proposed in the current paper does not need any background mesh, it is meshless and consequently independent of the geometry of domain. The error estimate and the convergence rate of the approach are studied for the presented method. The convergence accuracy of the new technique is examined over four integral equations on the tear, annular, crescent, and castle domains, and obtained results confirm the theoretical error estimates.

The numerical solution of nonlinear integral equations of the second kind using thin plate spline discrete collocation method

In this article, the thin plate splines are given to obtain the numerical solution of nonlinear Fredholm integral equations of the second kind. The scheme approximates the solution using the discrete collocation method based on the shape functions of thin plate splines constructed on a set of scattered points. The thin plate splines can be seen as a type of the free shape parameter radial basis functions which are used to establish an effective technique for the interpolation of an unknown function. The numerical method developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule to compute its integrals. Since the proposed scheme does not require any background mesh for approximations and numerical integrations, it is meshless. This approach can be easily implemented and its algorithm is simple and effective to solve nonlinear integral equations. Moreover the error estimate and the convergence rate of the method are presented. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.

A Least Squares Radial Basis Function Partition of Unity Method for Solving PDEs

Recently, collocation-based radial basis function (RBF) partition of unity methods (PUMs) for solving partial differential equations have been formulated and investigated numerically and theoretically. When combined with stable evaluation methods such as the RBF-QR method, high order convergence rates can be achieved and sustained under refinement. However, some numerical issues remain. The method is sensitive to the node layout, and condition numbers increase with the refinement level. Here, we propose a modified formulation based on least squares approximation. We show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling. We derive theoretical error estimates both for the collocation and least squares RBF-PUMs. Numerical experiments are performed for the Poisson equation in two and three space dimensions for regular and irregular geometries. The convergence experiments confirm the theoretical estimates, and the least squares formulation is shown to be 5--10 times faster than the collocation formulation for the same accuracy.

Multi-Dimensional Filtering: Reducing the Dimension Through Rotation

Over the past few decades there has been a strong effort toward the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters for discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this postprocessor. These advantages can be exploited during flow visualization, for example, by applying the SIAC filter to DG data before streamline computations [M. Steffen, S. Curtis, R. M. Kirby, and J. K. Ryan, IEEE Trans. Vis. Comput. Graphics, 14 (2008), pp. 680--692]. However, introducing these filters in engineering applications can be challenging since a tensor product filter grows in support size as the field dimension increases, becoming computationally expensive. As an alternative, [D. Walfisch, J. K. Ryan, R. M. Kirby, and R. Haimes, J. Sci. Comput., 38 (2009), pp. 164--184] proposed a univariate filter implemented along the streamline curves. Until now, this technique remained a numerical experiment. In this paper we introduce the line SIAC filter and explore how the orientation, structure, and filter size affect the order of accuracy and global errors. We present theoretical error estimates showing how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs, becoming an attractive tool for the scientific visualization community.

Numerical Methods for Power-Law Diffusion Problems

In this paper, we consider numerical methods for nonlinear diffusion problems where the diffusion term follows a power law, e.g., $p$-Laplace-type problems. In the first part, we present continuous higher order finite element discretizations for the model problem and we derive error estimates. In the second part, we discuss Newton iterative methods based on residual-based line-search and error-oriented globalization, which are employed for the numerical solution of the produced nonlinear algebraic system. Third, we formulate the original problem as a saddle point problem in the frame of augmented Lagrangian techniques and present two iterative methods for its solution. We conduct a systematic investigation of all solution algorithms. These algorithms are compared with respect to computational cost and their efficiency. Numerical results demonstrating the theoretical error estimates are also presented in five examples.

Approximation of Hilbert and Hadamard transforms on (0,+∞)

The authors propose a numerical method for computing Hilbert and Hadamard transforms on (0,+∞) by a simultaneous approximation process involving a suitable Lagrange polynomial of degree s and “truncated” Gaussian rule of order m, with s≪m. The proposed procedure is convergent and pointwise error estimates are given. Finally, some numerical tests confirming the theoretical error estimates are presented.

A new nonconforming finite element method for convection dominated diffusion-reaction equations

In this article, we introduce a non-conforming streamline diffusion finite element method for the approximation of convection dominated diffusion problems. The proposed non-conforming scheme is a hybrid type in which $$P_{1}$$ -nonconforming polynomials and $$P_{2}$$ -conforming polynomials on triangles are used. Computational advantages of the newly constructed element with rigorous error analysis for the streamline diffusion discretization is presented. Moreover, numerical experiments are conducted to demonstrate the efficiency of the proposed scheme and to validate the theoretical error estimates.

Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems

We propose a multiscale method based on a finite element heterogeneous multiscale method (in space) and the implicit Euler integrator (in time) to solve nonlinear monotone parabolic problems with multiple scales due to spatial heterogeneities varying rapidly at a microscopic scale. The multiscale method approximates the homogenized solution at computational cost independent of the small scale by performing numerical upscaling (coupling of macro and micro finite element methods). Taking into account the error due to time discretization as well as macro and micro spatial discretizations, the convergence of the method is proved in the general Lp(W1,p) setting. For p = 2, optimal convergence rates in the L2(H1) and C0(L2) norm are derived. Numerical experiments illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.

Convergence Analysis on Unstructured Meshes of a DDFV Method for Flow Problems with Full Neumann Boundary Conditions

A Discrete Duality Finite Volume (DDFV) method to solve on unstructured meshes the flow problems in anisotropic nonhomogeneous porous media with full Neumann boundary conditions is proposed in the present work. We start with the derivation of the discrete problem. A result of existence and uniqueness of a solution for that problem is given thanks to the properties of its associated matrix combined with adequate assumptions on data. Their theoretical properties, namely, stability and error estimates (in discrete energy norms andL2-norm), are investigated. Numerical test is provided.