Spectral collocation method for compact integral operators

We propose and analyze a spectral collocation method for integral equations with compact kernels, e.g. piecewise smooth kernels and weakly singular kernels of the form $\frac{1}{|t-s|^\mu}, \; 0\u3c\mu\u3c1.$ We prove that 1) for integral equations, the convergence rate depends on the smoothness of true solutions $y(t)$. If $y(t)$ satisfies condition (R): $\|y^{(k)}\|_{L^\infty[0,T]}\leq ck!R^{-k}$}, we obtain a geometric rate of convergence; if $y(t)$ satisfies condition (M): $\|y^{(k)}\|_{L^{\infty}[0,T]}\leq cM^k$, we obtain supergeometric rate of convergence for both Volterra equations and Fredholm equations and related integro differential equations; 2) for eigenvalue problems, the convergence rate depends on the smoothness of eigenfunctions. The same convergence rate for the largest modulus eigenvalue approximation can be obtained. Moreover, the convergence rate doubles for positive compact operators. Our numerical experiments confirm our theoretical results

Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with variable coefficient

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 Taylor & Francis.In this paper, a numerical implementation of a direct united boundary-domain integral equation (BDIE) related to the Neumann boundary value problem for a scalar elliptic partial differential equation with a variable coefficient is discussed. The BDIE is reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretization of the BDIEs with quadrilateral domain elements leads to a system of linear algebraic equations (discretized BDIE). Then, the system is solved by LU decomposition and Neumann iterations. Convergence of the iterative method is discussed in relation to the distribution of eigenvalues of the corresponding discrete operators calculated numerically.The work was supported by the grant EP/H020497/1 "Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients" of the EPSRC, UK

A Technique for Solving the Singular Integral Equations of Potential Theory

The singular integral equations of Potential Theory are investigated using ideas from both classical and contemporary mathematics. The goal of this semi-analytic approach is to produce numerical schemes that are both general and computationally simple. Previous works based on classical methods have yielded solutions only for very special cases while contemporary methods such as finite differences, finite elements and boundary element techniques are computationally extensive. Since the two-dimensional integral equations of interest exhibit structural invariance under a wide class of conformal mappings initial emphasis is placed on circular domains. By Fourier expansion with respect to the angular variable, such two-dimensional integral equations yield simultaneous systems of one-dimensional integral equations that, in many cases, uncouple. Integral transform techniques and classical function theory are used to identify the eigenfunctions associated with the dominant parts of the onedimensional singular equations. Hilbert spaces spanned by these eigenfunctions are then constructed and an operator theory developed for the general class of integral equations. Numerical algorithms are derived for both Galerkin and collocation solution techniques with convergence proved in the Galerkin case and collocation method verified experimentally. A generalization of the Hilbert space theory is then applied to the two-dimensional case with eigenfunctions created by combining the angular Fourier terms with the radial eigenfunctions of the dominant one-dimensional parts. Numerical algorithms based Galerkin and collocation methods are again derived and used to solve the two-dimensional equations. The techniques developed are used to solve a number of both previously known and new problems in Electrostatics and Fracture Mechanics. Simple layer potential representations yield weakly singular integral equations for the induced charge on disc shaped conductors that are placed in an electrostatic field. Similarly, double layer potentials yield hyper-singular integral equations for the crack opening displacement of penny shaped cracks in an elastic solid under various loading conditions. Conformal mapping techniques for solving problems on non-circular domains are also briefly discussed as are extensions to fields that are governed by the Helmholtz Equation

Transverse electric scattering on inhomogeneous objects: spectrum of integral operator and preconditioning

The domain integral equation method with its FFT-based matrix-vector products is a viable alternative to local methods in free-space scattering problems. However, it often suffers from the extremely slow convergence of iterative methods, especially in the transverse electric (TE) case with large or negative permittivity. We identify the nontrivial essential spectrum of the pertaining integral operator as partly responsible for this behavior, and the main reason why a normally efficient deflating preconditioner does not work. We solve this problem by applying an explicit multiplicative regularizing operator, which transforms the system to the form `identity plus compact', yet allows the resulting matrix-vector products to be carried out at the FFT speed. Such a regularized system is then further preconditioned by deflating an apparently stable set of eigenvalues with largest magnitudes, which results in a robust acceleration of the restarted GMRES under constraint memory conditions.Comment: 20 pages, 8 figure

A Personal Perspective on the History of the Numerical Analysis of Fredholm Integral Equations of the Second Kind

This is a personal perspective on the development of numerical methods for solving Fredholm integral equations of the second kind, discussing work being done principally during the 1950s and 1960s. The principal types of numerical methods being studied were projection methods (Galerkin, collocation) and Nyström methods. During the 1950s and 1960s, functional analysis became the framework for the analysis of numerical methods for solving integral equations, and this in‡uenced the questions being asked. This paper looks at the history of the analyses being done at that time.

Numerical Analysis of the Potential Formulation of the Volume Integral Equation for Electromagnetic Scattering

Peer reviewe

Accurate and efficient three-dimensional electrostatics analysis using singular boundary elements and Fast Fourier Transform on Multipole (FFTM)

Ph.DDOCTOR OF PHILOSOPH

Poloidal-toroidal decomposition in a finite cylinder. II. Discretization, regularization and validation

The Navier-Stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility. Regularity of the solutions is ensured in several ways: First, the potentials are represented using a spectral basis which is analytic at the cylindrical axis. Second, the non-physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile. Third, the nonlinear term is evaluated in such a way as to eliminate singularities. The resulting pseudo-spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated. Our solutions are shown to agree with exact polynomial solutions and with previous axisymmetric calculations of vortex breakdown and of nonaxisymmetric calculations of onset of helical spirals. Parallelization by azimuthal wavenumber is shown to be highly effective