Narrow Results

Pocklington's model consists in a one-dimensional integral equation relating the current at the surface of a straight finite wire to the tangential trace of an incident electromagnetic field. It is a simplification of the more usual single layer potential equation posed on a two-dimensional surface. We are interested in estimating the error between the solution of the exact integral equation and the solution of Pocklington's model. We address this problem for the model case of acoustics in a smooth geometry using results of asymptotic analysis.

In this paper, four three-node triangular finite element models which can readily be incorporated into the standard finite element program framework are devised via a multi-field variational functional for the bounded plane Helmholtz problem. In the models, boundary and domain fields are independently assumed. The former is constructed by nodal interpolation and the latter comprises nonsingular solutions of the Helmholtz equation. The equality of the two fields are enforced along the element boundary. Among the four devised models, the most accurate one is 1/3 to 1/2 less erroneous than the conventional single-field model in most examples.

We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field $u_0$ such that the spectrum of $D{u}_0$ is positive and bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends Grassin and Serre’s papers [Existence de solutions globales et régulières aux équations d’Euler pour un gaz parfait isentropique, C. R. Acad. Sci. Paris Sér. I 325 (1997) 721–726, 1997; Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J. 47 (1998) 1397–1432; Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier Grenoble 47 (1997) 139–159] dedicated to the compressible Euler system without coupling and with integer regularity exponents.

The Helmholtz equation with zero boundary conditions is important in the vibration of membranes, simply-supported plates, and electromagnetic and quantum waveguides. Trapped modes, with a lowered frequency and limited local effect, may exist for infinite strips with local bends or enlargements. Infinite strips with a cross bulge, with an S bend and with a U bend are studied using an efficient eigenfunction expansion and domain decomposition method. As the bulge become larger, more trapped modes with more complexity appear. Some geometries do not support trapped modes at all. These results are new.