A numerical minimization scheme for the complex Helmholtz equation. (English) Zbl 1272.65095

Richins, Russell B.; Dobson, David C.

ESAIM, Math. Model. Numer. Anal. 46, No. 1, 39-57 (2012).

Summary: We use the work of G. W. Milton, P. Seppecher, and G. Bouchitté [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465, No. 2102, 367–396 (2009; Zbl 1186.74044)] on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate the method with numerical experiments.

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A25	Electromagnetic theory, general
78M10	Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory

Keywords:

variational methods; Helmholtz equation; finite element methods; error bound; numerical experiments