Boundary integral approximation of a heat-diffusion problem in time-harmonic regime

In this paper we propose and analyse numerical methods for the approximation of the solution of Helmholtz transmission problems in the half plane. The problems we deal with arise from the study of some models in photothermal science. The solutions to the problem are represented as single layer potentials and an equivalent system of boundary integral equations is derived. We then give abstract necessary and sufficient conditions for convergence of Petrov–Galerkin discretizations of the boundary integral system and show for three different cases that these conditions are satisfied. We extend the results to other situations not related to thermal science and to non-smooth interfaces. Finally, we propose a simple full discretization of a Petrov–Galerkin scheme with periodic spline spaces and show some numerical experiments.     

Aubin—Nitsche Estimates Are Equivalent to Compact Embeddings

In this short note we show that having an Aubin–Nitsche type estimate (a superconvergence estimate in a weaker norm for a convergent Galerkin method) is equivalent to having compact injection of the space into its completion with the weaker norm.     

The Euler-Maclaurin Formula in Presence of a Logarithmic Singularity

In this short note we prove an extension of the Euler-Maclaurin expansion for general rectangular composite quadrature rules in one dimension when the derivative of the integrand has a logarithmic singularity. We show that a correction series has to be added to the formula, but that the asymptotic expansion in powers of the discretization parameter still holds.     

Author Index

Authors Index

Author Index     

Full asymptotics of spline Petrov–Galerkin methods for some periodic pseudodifferential equations

In this paper we study the existence of a formal series expansion of the error of spline Petrov–Galerkin methods applied to a class of periodic pseudodifferential equations. From this expansion we derive some new superconvergence results as well as alternative proofs of already known weak norm optimal convergence results. As part of the analysis the approximation of integrals of smooth functions multiplied by splines by rectangular rules is analyzed in detail. Finally, some numerical experiments are given to illustrate the applicability of Richardson extrapolation as a means of accelerating the convergence of the methods.     

The Quenching Set of a MEMS Capacitor in Two-Dimensional Geometries

The formation of finite time singularities in a nonlinear parabolic fourth order partial differential equation (PDE) is investigated for a variety of two-dimensional geometries. The PDE is a variant of a canonical model for Micro–Electro Mechanical systems (MEMS). The singularities are observed to form at specific points in the domain and correspond to solutions whose values remain finite but whose derivatives diverge as the finite time singularity is approached. This phenomenon is known as quenching. An asymptotic analysis reveals that the quenching set can be predicted by simple geometric considerations suggesting that the phenomenon described is generic to higher order parabolic equations which exhibit finite time singularity.     

Asymptotic expansions of the error of spline Galerkin boundary element methods

Summary. In this paper we analyse the existence of asymptotic expansions of the error of Galerkin methods with splines of arbitrary degree for the approximate solution of integral equations with logarithmic kernels. These expansions are obtained in terms of an interpolation operator and are useful for the application of Richardson extrapolation and for obtaining sharper error bounds. We also present and analyse a family of fully discrete spline Galerkin methods for the solution of the same equations. Following the analysis of Galerkin methods, we show the existence of asymptotic expansions of the error. Received May 18, 1995 / Revised version received April 11, 1996     

Mixed boundary integral methods for Helmholtz transmission problems

In this paper we propose a hybrid between direct and indirect boundary integral methods to solve a transmission problem for the Helmholtz equation in Lipschitz and smooth domains. We present an exhaustive abstract study of the numerical approximation of the resulting system of boundary integral equations by means of Galerkin methods. Some particular examples of convergent schemes in the smooth case in two dimensions are given. Finally, we extend the results to a thermal scattering problem in a half plane with several obstacles and provide numerical results that illustrate the accuracy of our methods depending on the regularity of the interface.