A meshless Galerkin method for Stokes problems using boundary integral equations

A meshless Galerkin scheme for the simulation of two-dimensional incompressible viscous fluid flows in primitive variables is described in this paper. This method combines a boundary integral formulation for the Stokes equation with the moving least-squares (MLS) approximations for construction of trial and test functions for Galerkin approximations. Unlike the domain-type method, this scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns, thus it is especially suitable for the exterior problems. Compared to other meshless methods such as the boundary node method and the element free Galerkin method, in which the MLS is also introduced, boundary conditions do not present any difficulty in using this meshless method. The convergence and error estimates of this approach are presented. Numerical examples are also given to show the efficiency of the method.     

On the error analysis of the numerical solution of linear Volterra integral equations of the second kind

In the numerical solution of linear Volterra integral equations, two kinds of errors occur. If we use the collocation method, these errors are the collocation and numerical quadrature errors. Each error has its own effect in the accuracy of the obtained numerical solution. In this study we obtain an error bound that is sum of these two errors and using this error bound the relation between the smoothness of the kernel in the equation and also the length of the integration interval and each of these two errors are considered. Concluded results also are observed during the solution of some numerical examples.     

Effective error estimates for the numerical solution of Fredholm integral equations

This paper describes the theoretical basis of an effective a posteriori error analysis for Fredholm integral equations of the second kind. We present error bounds from which we then derive error estimators that are easily computable, reliable, and in most cases reasonably precise.     

On the numerical solution of some algebraic integral equations

We discuss the numerical solution of some algebraic integral equations and integral equations of Lichtenstein type by means of a variational method and by using the subspace ofL-splines. For these approximate solutions the proofs of existence, uniqueness, and convergence as well as error estimates are given.     

On the numerical solution of Fredholm integral equations utilizing the local radial basis function method

The current investigation describes a computational technique to solve one- and two-dimensional Fredholm integral equations of the second kind. The method estimates the solution using the discrete collocation method by combining locally supported radial basis functions (RBFs) constructed on a small set of nodes instead of all points over the analysed domain. In this work, we employ the Gauss–Legendre integration rule on the influence domains of shape functions to approximate the local integrals appearing in the method. In comparison with the globally supported RBFs for solving integral equations, the proposed method is stable and uses much less computer memory. The scheme does not require any cell structures, so it is meshless. We also obtain the error analysis of the proposed method and demonstrate that the convergence rate of the approach is high. Illustrative examples clearly show the reliability and efficiency of the new method.     

An adaptive Huber method with local error control,for the numerical solution of the first kind Abel integral equations

In contrast to the existing plethora of adaptive numerical methods for differential and integro-differential equations, there seems to be a shortage of adaptive methods for purely integral equations with weakly singular kernels, such as the first kind Abel equation. In order to make up this deficiency, an adaptive procedure based on the product-integration method of Huber is developed in this work. In the procedure, an a posteriori estimate of the dominant expansion term of the local discretisation error at a given grid node is used to determine the size of the next integration step, in a way similar to the adaptive solvers for ordinary differential equations. Computational experiments indicate that in practice the control of the local errors is sufficient for bringing the true global errors down to the level of a prescribed error tolerance. The lower limit of the acceptable values of the error tolerance parameter depends on the interference of machine errors, and the quality of the approximations available for the method coefficients specific for a given kernel function.      

Approximate solution of systems of Volterra integral equations with error analysis

This paper describes a procedure for solving the system of linear Volterra integral equations by means of the Sinc collocation method. A convergence and an error analysis are given; it is shown that the Sinc solution produces an error of order O(exp(?c N ^1/2)), where c>0 is a constant. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The analytical results are illustrated with numerical examples that exhibit the exponential convergence rate.     

A meshless approximate solution of mixed Volterra–Fredholm integral equations

This paper presents a meshless method using a radial basis function collocation scheme for numerical solution of mixed Volterra–Fredholm integral equations, where the region of integration is a non-rectangular domain. We will show that this method requires only a scattered data of nodes in the domain. It is shown that the proposed scheme is simple and computationally attractive. Applications of the method are also demonstrated through illustrative examples.     

On the numerical solution of state equations

Two new methods for the solution of state equations of a linear time-invariant system are suggested. These methods are based on Romberg's algorithm and utilize the special form of the function to be integrated. The suggested methods are compared with existing ones.     

A method for the numerical solution of singular integral equations with logarithmic singularities

A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated     

On the approximate solution of singular integral equations

A method for obtaining the approximate solution of singular integral equations of the first and second kinds is suggested. The solution is represented in the form of power series with undetermined coefficients multiplied by a function in which the essential features of the singularity of the solution are preserved. The method of collocations is used to determine the unknown coefficients. The examples show that the method suggested is more general and gives good results even in the case when the form of solution does not exactly preserve the essential features of singularity. The method is simpler than others which use the properties of orthogonal polynomials, and is applicable for the solution of single equations as well as systems of simultaneous equations.     

On the first order local stability scheme for the numerical solution of time-dependent compressible flows

The diffusion, stability and monotonicity properties of the first order local stability scheme are investigated by resorting to both analysis and numerical experimentation. The scheme, deducible from either the first order Rusanov scheme or the first order van Leer scheme, is optimally stable. The magnitude of the truncation viscosity of the scheme is in between that of the Rusanov scheme and of the second order Lax-Wendroff scheme. The scheme does not, in general, yield monotonic profiles of shocks. A modified version of the local stability scheme (which preserves the local stability character at each point in the flow region and switches over to the van Leer scheme near shocks only) is proposed for practical application. Tests on one-dimensional problems show that the scheme achieves a better resolution of the flow features than is possible with the Rusanov scheme. The scheme is found to be capable of yielding a resolution which is almost equal to that of the second order schemes. Furthermore, it offers other advantages in that it resists nonlinear instabilities, is easy to programme and requires only a modest amount of storage and time on the computer.     

Higher order numerical solution of the integral equation for the two-dimensional neumann problem

Interest in the problem of two-dimensional potential flow in arbitrary multiply-connected domains has been stimulated by the need to calculate flow about multiple airfoil configurations consisting of slats and flaps detached from the main airfoil. General methods of solution are based on the use of a singularity distribution over the boundary. The distribution is obtained as the solution of an integral equation over the boundary. In implementing this solution various investigators approximate the boundary by an inscribed polygon, whose faces are small flat surface elements. The singularity on each element is taken as constant by some investigators and linearly varying by others. This paper systematically investigates the effectiveness of higher order approximations of the integral equation, including use of curved surface elements and parabolically-varying singularity. It is found that the approach using flat elements with constant singularity is mathematically consistent as is the next higher-order approach with parabolic elements and linearly varying singularity. The popular approach based on flat elements with linearly varying singularity is shown to be mathematically inconsistent, and examples are presented for which the effect of element curvature is greater than that of the singularity derivative. A number of examples are presented to show that: (1) the higher order solutions give very little increase in accuracy for the important case of exterior flow about a convex body: (2) for bodies with substantial concave regions and for interior flows in ducts, the use of parabolic elements and linearly varying singularity can give a dramatic increase in accuracy; and (3) the use of still higher order solutions leads to a rather small additional gain in accuracy.     

On the numerical solution of the stokes equations using the finite element method

A numerical solution of the stationary Stokes equations is considered based on the work of Crouzeix and Raviart [1]. The finite element method is used to discretize the partial differential equations, and a direct discretization of the velocity field and pressure is given which is applicable in both two and three dimensions. It is shown that not every arbitrary element can be used, and a condition is given to check whether or not an element is admissible. The system of linear equations is solved using the method of Powell and Hestenes for constrained optimization (see [2]).     

Adaptive analysis of plates and laminates using natural neighbor Galerkin meshless method

In this paper, natural neighbor Galerkin meshless method is employed for adaptive analysis of plates and laminates. The displacement field and strain field of plate are based on Reissner–Mindlin plate theory. The interpolation functions employed here were developed by Sibson and based on natural neighbor coordinates. An adaptive refinement strategy based on recovery energy norm which is in turn based on natural neighbors is employed for analysis of plates. The present adaptive procedure is applied to classical plate problems subjected to in-plane loads. In addition to that the laminated composite plates with cutouts subjected to transverse loads are investigated. Influence of the location of the cutout and the boundary conditions of the plate on the results have been studied. The results obtained with present adaptive analysis are accurate at lower computational effort when compare to that of no adaptivity. Further, the adaptive analysis provided accurate magnitude of maximum stresses and their locations in the laminate plates with and without cutout subjected to transverse loads. Additionally, failure prone areas in the geometry of the plates subjected to loads are revealed with the adaptive analysis.