On diagonal dominance of stiffness matrices in 3D. (English) Zbl 0824.65112

Křížek, M.; Qun, L.

East-West J. Numer. Math. 3, No. 1, 59-69 (1995).

A class of nonlinear elliptic boundary value problems on polyhedral domains which are approximated by standard linear tetrahedral elements are considered. The problems under consideration describe magnetic fields, nonviscous subsonic irrotational flows, nonlinear heat conduction in isotropic media, etc. As the continuous problem satisfies the maximum principle, the authors introduce an acute type condition upon the partitions of the domain which guarantees that the discrete problem satisfies the discrete maximum principle. It is proven that the corresponding stiffness matrix is irreducibly diagonally dominant if all internal angles between the faces of the tetrahedra are not greater than \(\pi/2\). Using this fact, the satisfaction of the discrete maximum principle is shown.
Some examples of acute type decompositions (all six internal angles between the faces of any tetrahedron are not greater than \(\pi/2\)) are presented.

Reviewer: K.Georgiev (Sofia)

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10	Iterative numerical methods for linear systems
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

finite element method; domain decomposition; nonlinear elliptic boundary value problems; linear tetrahedral elements; maximum principle; acute type condition; stiffness matrix; diagonally dominant