On the numerical solution of the Helmholtz equation at large wave numbers using exact controllability methods. Application to scattering. (English) Zbl 0795.65038

Bristeau, M. O.; Glowinski, R.; Periaux, J.

Quarteroni, Alfio (ed.) et al., Domain decomposition methods in science and engineering. The sixth international conference on domain decomposition, Como, Italy, June 15-19, 1992. Providence, RI: American Mathematical Society. Contemp. Math. 157, 399-419 (1994).

A numerical method for the solution of the Helmholtz equation based on a least squares/shooting method for the underlying wave equation is developed and applied to several test problems. The method is closely related to exact controllability and to the Hilbert uniqueness method of J. L. Lions. A conjugate gradient algorithm with good convergence properties is described and the efficiency of the new algorithm is demonstrated on the example of scattering plane waves.
For the entire collection see [Zbl 0785.00036].

Reviewer: Th.Sonar (Göttingen)

MSC:

65K10	Numerical optimization and variational techniques
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
93C20	Control/observation systems governed by partial differential equations
93B05	Controllability
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

large wave numbers; optimal control problems; Helmholtz equation; least squares/shooting method; wave equation; test problems; exact controllability; conjugate gradient algorithm; convergence; algorithm; scattering plane waves