Multiscalet basis in Galerkin's method for solving three‐dimensional electromagnetic integral equations

This paper briefly describes the development of generalized collocation methods for solving coupled systems of nonlinear, one-dimensional, parabolic, partial differential equations. parabolic, partial differential equations. The methods are applied to several problems that are representative of fluid flow in a porous medium. The numerical results for the problems solved indicate that collocation methods using C piecewise polynomials with Gauss-Legendre collocation points polynomials with Gauss-Legendre collocation points are more efficient than conventional, second-order, finite-difference methods. In particular, for problems whose solutions are smooth, one obtains more accuracy per unit of computer time with increasingly higher-order collocation methods. For problems whose solutions are characterized by a steep, shock-like, transient wave front, one can conclude that increasing the order of the collocation method does not necessarily result in a more efficient method, but high-accuracy collocation methods still appear to be more efficient than conventional, second-order, finite-difference methods. Also, for a given accuracy, collocation methods generally require less computer storage than conventional, second-order, finite-difference methods. Finally, the collocation technique presented here is shown to be not directly applicable to the saturation equation for immiscible flow. Introduction The purpose of this paper is to investigate collocation methods for the numerical solution of nonlinear equations similar to those that describe the flow of a fully compressible fluid in a porous medium and those that describe the process by which one miscible liquid displaces another liquid in a porous medium. The objective is to determine if collocation methods are applicable and economically justified for solving these types of petroleum engineering problems. The conclusions petroleum engineering problems. The conclusions reached are based on the amount of computer time and computer storage that is required to obtain a specified accuracy. Since only two problems are solved, the results may not be extendable to more general problems; however, experience in solving a wide variety of problems indicates that this is the general behavior that might be expected when using collocation methods. Previously published results indicate how Galerkin, collocation, and finite-element methods can be used to solve various problems that arise in petroleum engineering. Most of these papers present a variety of arguments in favor of these present a variety of arguments in favor of these methods. The principal argument is that better answers can be obtained for the same computational effort than by finite-difference methods. Culham and Varga have presented the most convincing evidence in favor of Galerkin methods. However, they only considered piecewise linear and piecewise cubic-basis functions and they only considered one problem. Because of these limitations, it is not problem. Because of these limitations, it is not clear whether their conclusions can be extended to higher-order Galerkin methods or to other types of problems. Also, they did not consider collocation problems. Also, they did not consider collocation methods in any detail. This paper considers the use of increasingly higher-order collocation methods and compares the results with those obtained with the usual second-order, finite-difference method. The Galerkin method is not considered, even though that is the method with which collocation must ultimately compete. This was done for several reasons. A preliminary operation count indicates that Galerkin methods will not be competitive with collocation methods for most problems. Also, it is not clear how to implement the higher-order Galerkin methods to achieve maximum efficiency. These remarks indicate that the relative merits of collocation and Galerkin methods are still uncertain. PROBLEMS SOLVED PROBLEMS SOLVED Two problems are solved to serve as a basis for the comparisons. The first problem, called the "gas flow problem," is the same problem considered by Culham and Varga. This problem is nonlinear and includes a volumetric source term. SPEJ P. 345

In the previous chapters, with an exception of GPG, the finite element formulations are based on the Galerkin methods in which test functions are chosen to be the same as the trial functions. This is not required in the weighted residual methods. Weighted residual methods other than the Galerkin methods include spectral element methods (SEM), least square methods (LSM), moment methods, or collocation methods, in which the test functions or weighting functions are not necessarily the same as the trial functions. In spectral element methods (SEM), polynomials in terms of nodal values of the variables are combined with special functions such as Chebyshev or Legendre polynomials. For least square methods, the test functions are constructed by the derivative of the residual with respect to the nodal values of the variables. Some arbitrary functions are chosen as test functions for the moment and collocation methods. Recently, the weighted residual concept has been used in meshless configurations, known as the finite point method (FPM), partition of unity method, meshless cloud method, or element-free method.

The Numerical Solution of Integral Equations on the Half-Line

Numerical experiments involving galerkin and collocation methods for linear integral equations of the first kind

Spectral methods are evaluated for the solution of population balance problems. Both a simplified population balance equation (PBE) with an analytical solution available and a rigorous PBE are considered in this numerical analysis. In comparison with the orthogonal collocation, tau, and Galerkin methods, the least-squares method does not produce the same favorable results. On the other hand, the least-squares method with a direct minimization solver is capable of producing equally accurate results of the model equations as the orthogonal collocation, tau, and Galerkin methods. Compared to the conventional least-squares method, the direct minimization formulation is better conditioned but does not produce a symmetric positive-definite system matrix.

We consider the method of evaluating matrix elements of electromagnetic volume integral equations with the help of suitably constructed Laplacian-type representations of singular kernels (Green functions) appearing in electromagnetic volume and surface integral equations. The method consists of representing the singular kernels (Green functions) in terms of a generalized Laplacian operator acting on some auxiliary functions. Such a representations allow us, by using Gauss divergence theorem, to convert volumetric and surface integrals representing matrix elements to integrals always involving only non-singular integrands. The task of of finding particular Laplacian representation of different kernels amounts to solving appropriate ordinary or partial inhomogeneous differential equations. We apply the Laplacian method to analytic evaluation of matrix elements of the volume integral equation by employing two different Laplacian representations of the integral equation kernels. The first representation allows us to reduce 6-dimensional volume integrals over tetrahedra to 2-dimensional surface integrals over tetrahedra faces. The second representation is subsequently used reducing surface integrals over tetrahedra faces to pairs of of 1-dimensional line integrals over tetrahedra edges. We also derive analytic expressions for the resulting line integrals given in terms of elementary functions.

This study presents a rigorous and in-depth comparative analysis of the Chebyshev Collocation and Spectral Galerkin methods for solving nonlinear singular initial value problems, which are fundamental in modeling astrophysical phenomena, with a focus on the Lane-Emden equation as a model test case, by applying both methodologies, performance was evaluated based on precise metrics including accuracy, rate of convergence, computational efficiency, the results unequivocally demonstrate the superiority of the Spectral Galerkin method across all critical performance aspects, the method achieved true exponential convergence for smooth solutions, reaching a precision approaching machine accuracy using significantly lower approximation degrees than the collocation method. Furthermore, the Galerkin method exhibited superior robustness and a higher algebraic rate of convergence in cases where the solution is less regular, while maintaining excellent numerical stability and better-conditioned algebraic systems, we conclude that the integral nature of the orthogonal projection in the Galerkin method endows it with a structural advantage

This comprehensive resource conveniently combines advanced topics of electromagnetic theory, a high level of mathematical detail, and the well-established ubiquitous Method of Moments applied to the solution of practical wave-scattering and antenna problems formulated with surface, volume, and hybrid integral equations

Many mathematical problems in science and engineering are defined by ordinary or partial differential equations with appropriate initial-boundary conditions. Among the various methods, boundary integral equation method (BIEM) is probably the most effective. It’s main advantage is that it changes a problem from its formulation in terms of unbounded differential operator to one for an integral/integro-differential operator, which makes the problem tractable from the analytical or numerical point of view. Basically, the review/study of the problem is shifted to a boundary (a relatively smaller domain), where it gives rise to integral equations defined over a suitable function space. Integral equations with singular kernels areamong the most important classes in the fields of elasticity, fluid mechanics, electromagnetics and other domains in applied science and engineering. With the advancesin computer technology, numerical simulations have become important tools in science and engineering. Several methods have been developed in numerical analysis for equations in mathematical models of applied sciences. Widely used methods include: Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM) and Galerkin Method (GM). Unfortunately, none of these are versatile. Each has merits and limitations. For example, the widely used FDM and FEM suffers from difficulties in problem solving when rapid changes appear in singularities. Even with the modern computing machines, analysis of shock-wave or crack propagations in three dimensional solids by the existing classical numerical schemes is challenging (computational time/memory requirements). Therefore, with the availability of faster computing machines, research into the development of new efficient schemes for approximate solutions/numerical simulations is an ongoing parallel activity. Numerical methods based on wavelet basis (multiresolution analysis) may be regarded as a confluence of widely used numerical schemes based on Finite Difference Method, Finite Element Method, Galerkin Method, etc. The objective of this monograph is to deal with numerical techniques to obtain (multiscale) approximate solutions in wavelet basis of different types of integral equations with kernels involving varieties of singularities appearing in the field of elasticity, fluid mechanics, electromagnetics and many other domains in applied science and engineering.

An effective numerical method based on wavelet matrix transforms for efficient solution of electromagnetic (EM) integral equations is proposed. Using the wavelet matrix transform produces highly sparse moment matrices which can be solved efficiently. A fast construction method for various orthonormal or nonorthonormal wavelet basis matrices is also given. It has been found that using nonsimilarity wavelet matrix transforms such as nonsimilarity nonorthonormal cardinal spline wavelet (NSNCSW) transform, one can obtain a much higher compression rate and much better accuracy of the approximate solutions than using similarity wavelet transforms such as Daubechies' (1992) orthonormal wavelet (DOW) transform. Numerical examples are given to show the validity and effectiveness of the method.

A general theory is given for discretized versions of the Galerkin method for solving Fredholm integral equations of the second kind.The discretized Galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the Galerkin method.The theoretical framework that is given parallels that of the regular Galerkin method, including the error analysis of the superconvergence of the iterated Galerkin and discrete Galerkin solutions.In some cases, the iterated discrete Galerkin solution is shown to coincide with the Nyström solution with the same numerical integration method.The paper concludes with applications to finite element Galerkin methods.1. Introduction.Consider the numerical solution of the Fredholm integral equation of the second kind, (1.1)x(s)-f K(s,t)x(t)da(t)=y(s), s e D. JDIn this paper we will define and analyze the discrete Galerkin method for the numerical solution of (1.1).The Galerkin method is a well-known procedure for the approximate solution of this equation (e.g., see [5, p. 62]); and the discrete Galerkin method results when the integrations of the Galerkin method are evaluated numerically.Before giving a more precise definition of the discrete Galerkin method, we review results for the Galerkin method.In Eq. (1.1), the region D is to be a closed subset of Rm, some m > 1; and the dimension of D can be less than m, for example, if D is a surface in R3.For the discrete Galerkin method, we will assume that K(s, t) is continuous for s, t e D, although that is not necessary for the discussion of Galerkin's method given below.The equation (1.1) is written symbolically as (1.2) (I-Jf)x=y, with the integral operator assumed to be a compact operator from L2(D) to L2(D) and from L°°(D) to C(D).Further, it is assumed that (1.1) is uniquely solvable in SC for all y e 3C, for both 3C= C(D) and SC= L2(D).Additional assumptions on D and K(s, t) will be given as they are needed in the applications presented later in the paper.Generally y e C(D), and this will imply x e C(D).

A discontinuous Petrov–Galerkin methodology for adaptive solutions to the incompressible Navier–Stokes equations

A comparison of techniques for local stability analysis of tubular reactor systems

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

The Laplace equation was solved in the interelectrode space for shaped electrodes (two-dimensional case) by the method of finite differences (FDM), Galerkin's method (GM), and collocation method (CM). A comparison shows that for electrode shapes with a continuously changing surface (continuous first and second derivatives), the solutions by all three methods are equivalent, giving identical distribution of local current densities on the electrode surface. The use of GM and CM is, however, not practical because of high requirements on the computer time and memory as compared with FDM. Moreover, the GM and CM fail in the case of discontinuities in the electrode shape.