Edge singularities in heat theory and solution of some hyperbolic problems. (Singularités d’arêtes en thermique et résolution de quelques problèmes hyperboliques.) (French) Zbl 0851.35007

Rocquencourt: Institut National de Recherche en Informatique et en Automatique (INRIA). x, 207 p. (Thèse, Univ. de Nice-Sophia-Antipolis) (1990).
The first part is concerned with the mathematical and numerical study of the boundary value (Neumann or Dirichlet) problem for the Poisson equation in a bounded three-dimensional domain with edges on its boundary. Two types of domains are considered: a cylinder with a nonconvex and polygonal basis, and an axisymmetric domain with a nonconvex and polygonal generating plane. In both cases, it is shown that the unique solution may be decomposed into a regular part (belonging to a Sobolev space) and a singular part obtained explicitly as a convolution product with a Poisson-type kernel multiplied by \(r^\beta\sin(\beta\theta)\). Here, \((r, \theta)\) are polar coordinates and \(\beta= \Pi/\omega\), where \(\omega\) measures the nonconvex sector of the boundary. The numerical approximation combines Fourier series expansion (in one space variable) and finite element method (in the two other space variables). It is shown that the above-mentioned decomposition may be used to improve the accuracy of the approximation. Moreover, error estimates (in Sobolev norm) are proved.
The second part is devoted to the study of numerical schemes for several hyperbolic equations. First of all, a family of third-order schemes (in space and time) is constructed and tested numerically for the advection equation (1D and 2D) and for the Euler system (1D). Next, the author is concerned with the Euler system (3D) in a bounded axisymmetric domain.
It is shown that a numerical approximation in the generating section of the domain may be obtained by first constructing a finite volume approximation in the 3D domain, and then passing to the limit with respect to a small angle. The scheme thus obtained is analyzed (modified equation, dissipation and dispersion) on a linear advection equation. Finally, several multigrid algorithms are described and compared when used to solve the Euler (2D) system.


35A35 Theoretical approximation in context of PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs