Universal quadratures for boundary integral equations on two-dimensional domains with corners

We describe the construction of a collection of quadrature formulae suitable for the efficient discretization of certain boundary integral equations on a very general class of two-dimensional domains with corner points. The resulting quadrature rules allow for the rapid high-accuracy solution of Dirichlet boundary value problems for Laplace’s equation and the Helmholtz equation on such domains under a mild assumption on the boundary data. Our approach can be adapted to other boundary value problems and certain aspects of our scheme generalize to the case of surfaces with singularities in three dimensions. The performance of the quadrature rules is illustrated with several numerical examples.     

Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations

For optical waveguides with high index-contrast and sharp corners, existing full-vectorial mode solvers including those based on boundary integral equations typically have only second or third order of accuracy. In this paper, a new full-vectorial waveguide mode solver is developed based on a new formulation of boundary integral equations and the so-called Neumann-to-Dirichlet operators for sub-domains of constant refractive index. The method uses the normal derivatives of the two transverse magnetic field components as the basic unknown functions, and it offers higher order of accuracy where the order depends on a parameter used in a graded mesh for handling the corners. The method relies on a standard Nyström method for discretizing integral operators and it does not require analytic properties of the electromagnetic field (which are singular) at the corners.     

Efficient solution strategy for the semi-implicit discontinuous Galerkin discretization of the Navier-Stokes equations

We deal with the numerical solution of the system of the compressible Navier–Stokes equations with the aid of the interior penalty Galerkin method. We employ a semi-implicit time discretization which leads to the solution of a sequence of linear algebraic systems. We develop an efficient strategy for the solution of these systems. It is based on a simple adaptive technique for the choice of the time step and a relatively weak stopping criterion for iterative linear algebraic solvers. The presented numerical experiments show that the proposed strategy is efficient for steady-state problems using various grids, polynomial degrees of approximations and flow regimes. Finally, we apply this strategy with a minor modification to an unsteady flow.     

Sound radiation by baffled plates and related boundary integral equations

The displacement of the plate is described by its Green's representation, involving the infinite fluid-loaded plate kernel. The boundary conditions lead to a system of two integral equations along the plate boundary. From a numerical point of view, the method is efficient because this kernel is represented in terms of Hankel functions and Laplace type integrals, which can be computed very fast. Numerical methods are described and an example is given.     

Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps

Efficient numerical methods for analyzing photonic crystals (PhCs) can be developed using the Dirichlet-to-Neumann (DtN) maps of the unit cells. The DtN map is an operator that takes the wave field on the boundary of a unit cell to its normal derivative. In frequency domain calculations for band structures and transmission spectra of finite PhCs, the DtN maps allow us to reduce the computation to the boundaries of the unit cells. For two-dimensional (2D) PhCs with unit cells containing circular cylinders, the DtN maps can be constructed from analytic solutions (the cylindrical waves). In this paper, we develop a boundary integral equation method for computing DtN maps of general unit cells containing cylinders with arbitrary cross sections. The DtN map method is used to analyze band structures for 2D PhCs with elliptic and other cylinders.     

Ultraboundedness for parabolic equations in convex domains without boundary conditions

We consider the operator , where U is a convex real function defined in a convex open set O⊂R^N and lim_|x|→∞U(x)=lim_x→∂OU(x)=+∞. We prove that the associated Markov semigroup is ultrabounded with respect to the Gibbs measure .     

A note on discretization of nonlinear differential equations

An important issue when integrating nonlinear differential equations on a digital computer is the choice of the time increment or step size. For example, it is known that if this quantity is not sufficiently short, spurious chaotic motions may be induced when integrating a system using several of the well-known methods available in the literature. In this paper, a new approach to discretize differential equations is analyzed in light of computational chaos. It will be shown that the fixed points of the continuous system are preserved under the new discretization approach and that the spurious fixed points generated by higher order approximations depend upon the increment parameter. (c) 2002 American Institute of Physics.     

On the discretization of Laine equations

We consider the discretization of Darboux integrable equations. For each of the integrals of a Laine equation we constructed either a semi-discrete equation which has that integral as an n-integral, or we proved that such an equation does not exist. It is also shown that all constructed semi-discrete equations are Darboux integrable.     

Behavior of solutions of semilinear elliptic equations in domains with complicated boundary

In the paper, boundary value problems in a bounded domain for semilinear elliptic equations are considered. It is claimed that, for small exponent of nonlinearity, the support of a bounded solution is concentrated near the part of the boundary of the domain where the boundary condition is inhomogeneous. Estimates for the size of a neighborhood containing the support of the solution are established. For a supercritical exponent of nonlinearity, the convergence of solutions to a limit solution in the unperforated domain is established for some family of perforated domains. The rate of convergence is polynomial, and it depends on the exponent of the nonlinearity and on the rate at which the sizes of the cavities decrease simultaneously with the growth of the number of cavities and does not depend on the boundary conditions at the boundaries of the cavities. No restrictions on the displacement of the cavities are imposed.     

Efficient absorbing boundary conditions for Biot's equations in time-harmonic finite element applications

Absorbing boundary conditions for two phase media previously presented by Zerfa and Loret [Earthquake Eng. Struct. Dyn. 33, 89-110 (2004)] have been improved by considering additionally absorbing waves with auxiliary angles of incidence. These angles are defined at each point on the boundaries, so one can easily implement tensor impedances as analogous to those defined by Krenk and Kirkegaard for isotropic, nonporous media [J. Sound Vib. 247, 875-896 (2001)]. The boundary conditions have been tested and validated in two-dimensional frequency domain simulations.     

Laplace and the era of differential equations

Between about 1790 and 1850 French mathematicians dominated not only mathematics, but also all other sciences. The belief that a particular physical phenomenon has to correspond to a single differential equation originates from the enormous influence Laplace and his contemporary compatriots had in all European learned circles. It will be shown that at the beginning of the nineteenth century Newton's “fluxionary calculus” finally gave way to a French-type notation of handling differential equations. A heated dispute in the Philosophical Magazine between Challis, Airy and Stokes, all three of them famous Cambridge professors of mathematics, then serves to illustrate the era of differential equations. A remark about Schrödinger and his equation for the hydrogen atom finally will lead back to present times.     

On the existence of solutions of the first boundary value problem for elliptic equations on unbounded domains

The problem mentioned in the title is studied.     

Regularization of boundary integral equations in the problems of wave field diffraction by curved boundaries

A regularization of the exact Fredholm integral equations for the field or its derivative on a scattering surface is proposed. This approach allows one to calculate the scattering or diffraction of pulsed wave fields by curved surfaces of arbitrary geometry. Mathematically, the method is based on the replacement of the exact Fredholm integral equations by their truncated analogs, in which the contributions of the geometrically shadowed regions are cancelled. This approach has a clear physical meaning and provides stable solutions even when the direct numerical solution of mathematically exact initial integral equations leads to unstable results. The method is mathematically substantiated and tested using the problem of plane-wave scattering by a cylinder as an example.     

Existence of solutions for sequential fractional differential equations with four-point nonlocal fractional integral boundary conditions

This paper investigates the existence of solutions for a nonlinear boundary value problem of sequential fractional differential equations with four-point nonlocal Riemann-Liouville type fractional integral boundary conditions. We apply Banach’s contraction principle and Krasnoselskii’s fixed point theorem to establish the existence of results. Some illustrative examples are also presented.     

Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes

We developed a new monotone finite volume method for diffusion equations. The second-order linear methods, such as the multipoint flux approximation, mixed finite element and mimetic finite difference methods, are not monotone on strongly anisotropic meshes or for diffusion problems with strongly anisotropic coefficients. The finite volume (FV) method with linear two-point flux approximation is monotone but not even first-order accurate in these cases. The developed monotone method is based on a nonlinear two-point flux approximation. It does not require any interpolation scheme and thus differs from other nonlinear finite volume methods based on a two-point flux approximation. The second-order convergence rate is verified with numerical experiments.     

Exact boundary condition for time-dependent wave equation based on boundary integral

An exact non-reflecting boundary conditions based on a boundary integral equation or a modified Kirchhoff-type formula is derived for exterior three-dimensional wave equations. The Kirchhoff-type non-reflecting boundary condition is originally proposed by L. Ting and M.J. Miksis [J. Acoust. Soc. Am. 80 (1986) 1825] and numerically tested by D. Givoli and D. Cohen [J. Comput. Phys. 117 (1995) 102] for a spherically symmetric problem. The computational advantage of Ting–Miksis boundary condition is that its temporal non-locality is limited to a fixed amount of past information. However, a long-time instability is exhibited in testing numerical solutions by using a standard non-dissipative finite-difference scheme. The main purpose of this work is to present a new exact boundary condition and to eliminate the long-time instability. The proposed exact boundary condition can be considered as a limit case of Ting–Miksis boundary condition when the two artificial boundaries used in their method approach each other. Our boundary condition is actually a boundary integral equation on a single artificial boundary for wave equations, which is to be solved in conjunction with the interior wave equation. The new boundary condition needs only one artificial boundary, which can be of any shape, i.e., sphere, cubic surface, etc. It keeps all merits of the original Kirchhoff boundary condition such as restricting the temporal non-locality, free of numerical evaluation of any special functions and so on. Numerical approximation to the artificial boundary condition on cubic surface is derived and three-dimensional numerical tests are carried out on the cubic computational domain.     

Positive solutions for a coupled system of semipositone fractional differential equations with the integral boundary conditions

In this paper, we investigate the existence of positive solutions for a system of nonlinear fractional differential equations with sign-changing nonlinearities. The result obtained in this paper essentially improves and extends some well-known results. An example demonstrates the main results.