RESUMEN

A two-step hybrid perturbation Galerkin technique for solving a variety of applied mathematics problems involving a small parameter is presented. The first step consists of using a regular or singular perturbation method to determine the asymptotic expansion of the solution in terms of the small parameter. Then the approximate solution is assumed to have the form of a sum of perturbation coefficient functions multiplied by (unknown) amplitudes (gauge functions). In the second step the classical Bubnov–Galerkin method is used to determine these amplitudes. The resulting hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Bubnov–Galerkin methods applied separately, while combining some of the good features of both. The proposed method is applied to some singular perturbation problems in slender body theory. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.

RESUMEN

It is well known that in the case of a regular domain the solution of the time-harmonic Maxwell's equations allows a discretization by means of nodal finite elements: this is achieved by solving a regularized problem similar to the vector Helmholtz equation. The present paper deals with the same problem in the case of a nonconvex polyhedron. It is shown that a nodal finite element method does not approximate in general the solution to Maxwell's equations, but actually the solution to a neighboring variational problem involving a different function space. Indeed, the solution to Maxwell's equations presents singularities near the edges and corners of the domain that cannot be approximated by Lagrange finite elements.

A new method is proposed involving the decomposition of the solution field into a regular part that can be treated numerically by nodal finite elements and a singular part that has to be taken into account explicitly. This singular field method is presented in various situations such as electric and magnetic boundary conditions, inhomogeneous media, and regions with screens.