Convergence of the Galerkin methods for equations with elliptic operators on subspaces and solving the electric field equation

Application of the Galerkin methods to the numerical analysis of the integro-differential electric field equation is justified. The convergence of the Galerkin methods is established for a class of equations with nonelliptic operators comprising the electric field equation. Theorems concerning the approximation of the elements belonging to a special Sobolev space by the basis Rao-Wilton-Glisson functions are proved. The rate of convergence is estimated.     

An exponential transformation based splitting method for fast computations of highly oscillatory solutions

Splitting, or decomposition, methods have been widely used for achieving higher computational efficiency in solving wave equations. A major concern has remained, however, if the wave number involved is exceptionally large. In the case, merits of a conventional splitting method may diminish due to the fact that tiny discretization steps need to be employed to compensate high oscillations. This paper studies an alternative way for solving highly oscillatory paraxial wave problems via a modified splitting strategy. In the process, an exponential transformation is first introduced to convert the underlying differential equation to coupled nonlinear equations. Then the resulted oscillation-free system is treated by a Local-One-Dimensional (LOD) scheme for desired accuracy, efficiency and computability. The splitting method acquired is asymptotically stable and easy to use. Computational experiments are given to illustrate our numerical procedures.     

On the convergence of the Galerkin method for nonsmooth solutions of integral equations

Summary It is shown that the stability region of the Galerkin method includes solutions not lying in the conventional energy space. Optimal order error estimates for these nonsmooth solutions are derived. The new result is compared with the classical statement by means of the basic potential problem.     

Mathematical analysis of a finite element method without spurious solutions for computation of dielectric waveguides

Summary In this paper we propose a finite element method to compute dielectric waveguides which does not present spurious solutions. In addition to a theoretical analysis showing the good stability properties of this method, we give numerical results for some classical test problems.This work was partially supported by the Accion Integrada Hispano-Francesa n. 259A, DGICYT, Spain     

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type (x−y)ln|x−y| and (x−y)²ln|x−y|. In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected.Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.     

Preconditioning techniques for the iterative solution of scattering problems

We consider a time-harmonic electromagnetic scattering problem for an inhomogeneous medium. Some symmetry hypotheses on the refractive index of the medium and on the electromagnetic fields allow to reduce this problem to a two-dimensional scattering problem. This boundary value problem is defined on an unbounded domain, so its numerical solution cannot be obtained by a straightforward application of usual methods, such as for example finite difference methods, and finite element methods. A possible way to overcome this difficulty is given by an equivalent integral formulation of this problem, where the scattered field can be computed from the solution of a Fredholm integral equation of second kind. The numerical approximation of this problem usually produces large dense linear systems. We consider usual iterative methods for the solution of such linear systems, and we study some preconditioning techniques to improve the efficiency of these methods. We show some numerical results obtained with two well known Krylov subspace methods, i.e., Bi-CGSTAB and GMRES.     

Multi-parameter error resolution for the collocation method of Volterra integral equations

In this paper, a multi-parameter error resolution technique is introduced and applied to the collocation method for Volterra integral equations. By using this technique, an approximation of higher accuracy is obtained by using a multi-processor in parallel. Additionally, a correction scheme for approximation of higher accuracy and a global superconvergence result are presented.     

Convergence of Galerkin approximations for the Kuramoto‐Tsuzuki equation

Standard Galerkin approximations, using smooth splines to solutions of the Kuramoto‐Tsuzuki equation are analyzed. The existence, uniqueness, and convergence of the fully discrete Crank‐Nicolson scheme are discussed. Furthermore, a second‐order convergent linearized Galerkin approximation are derived. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 21, 2005     

A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel

This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. Owing to the singular behavior of the solution near the origin, the global convergence order of product integration and collocation methods is not optimal. In order to recover the optimal orders a hybrid collocation method is used which combines a non-polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. A comparison is made with the standard graded collocation method.     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

Asymptotic error expansion for the Nyström method for nonlinear Fredholm integral equations of the second kind

In this paper the asymptotic error expansion for the Nyström method for one-dimensional nonlinear Fredholm integral equations of the second kind is considered. We show that the Nyström solution admits an error expansion in powers of the step-sizeh. Thus Richardson's extrapolation can be performed on the solution, and this will greatly increase the accuracy of the numerical solution.The project has been supported by the National Natural Science Foundation of China.     

Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations

In this paper we present local a-posteriori error indicators for the Galerkin discretization of boundary integral equations. These error indicators are introduced and investigated by Babuška-Rheinboldt [3] for finite element methods. We transfer them from finite element methods onto boundary element methods and show that they are reliable and efficient for a wide class of integral operators under relatively weak assumptions. These local error indicators are based on the computable residual and can be used for controlling the adaptive mesh refinement. Received March 4, 1996 / Revised version received September 25, 1996     

On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation

Summary We describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single-layer approach to the Dirichlet problem for the two-dimensional Helmholtz equation in smooth domains. We develop an error analysis in a Sobolev space setting and prove fast convergence rates for smooth boundary data.     

The Galerkin scheme for Lavrentiev’sm-times iterated method to solve linear accretive Volterra integral equations of the first kind

In this paper, for the numerical solution of linear accretive Volterra integral equations of the first kind in Hilbert spaces we consider the Galerkin scheme for Lavrentiev’sm-times iterated method, i.e., for each parameter choice for Lavrentiev’sm-times iterated method the arisingm stabilized equations are discretized by the Galerkin scheme. An associated discrepancy principle as parameter choice strategy for this finite-dimensional version of Lavrentiev’sm-times iterated method is proposed, and corresponding convergence results are provided.     

An integral equation method for the electromagnetic scattering from a scatterer on an absorbing plane

Consider a time-harmonic electromagnetic plane wave incident on a scatterer on a grounded absorbing plane modelized as an infinite impedance plane. In this paper, a new integral representation formula is rigorously derived. Existence and uniqueness of weak solutions for the model problem are also established. The proof of existence is based on an extension of the Hodge decomposition technique to open boundaries. The results reported in this paper form a basis for numerical solutions of the electromagnetic scattering problem from a scatterer on an absorbing plane.     

Legendre spectral collocation method for volterra-hammerstein integral equation of the second kind

This paper is concerned with obtaining the approximate solution for VolterraHammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function ω(x) = 1 as the collocation points. The Legendre collocation discretization is proposed for Volterra-Hammerstein integral equation. We provide an error analysis which justifies that the errors of approximate solution decay exponentially in L~2 norm and L~∞ norm. We give two numerical examples in order to illustrate the validity of the proposed Legendre spectral collocation method.     

Convergence of Galerkin solutions and continuous dependence on data in spectrally-hyperviscous models of 3D turbulent flow

We obtain results on the convergence of Galerkin solutions and continuous dependence on data for the spectrally-hyperviscous Navier-Stokes equations. Let uN denote the Galerkin approximates to the solution u, and let wN=u−uN. Then our main result uses the decomposition wN=PnwN+QnwN where (for fixed n) Pn is the projection onto the first n eigenspaces of A=−Δ and Qn=I−Pn. For assumptions on n that compare well with those in related previous results, the convergence of ‖QnwN(t)Hβ_‖ as N→∞ depends linearly on key parameters (and on negative powers of λn), thus reflective of Kolmogorov-theory predictions that in high wavenumber modes viscous (i.e. linear) effects dominate. Meanwhile ‖PnwN(t)Hβ_‖ satisfies a more standard exponential estimate, with positive, but fractional, dependence on λn. Modifications of these estimates demonstrate continuous dependence on data.