Nonconforming Discretization Research Articles

Multilevel preconditioners for discretizations of the biharmonic equation by rectangular finite elements

AbstractRecently, some new multilevel preconditioners for solving elliptic finite element discretizations by iterative methods have been proposed. They are based on appropriate splittings of the finite element spaces under consideration, and may be analyzed within the framework of additive Schwarz schemes. In this paper we discuss some multilevel methods for discretizations of the fourth‐order biharmonic problem by rectangular elements and derive optimal estimates for the condition numbers of the preconditioned linear systems. For the Bogner–Fox–Schmit rectangle, the generalization of the Bramble–Pasciak–Xu method is discussed. As a byproduct, an optimal multilevel preconditioner for nonconforming discretizations by Adini elements is also derived.

Finite Element Vibration Analysis of Fluid–Solid Systems without Spurious Modes

This paper deals with the finite element approximation of the vibration modes of a problem with fluid–structure interaction. Displacement variables are used for both the fluid and the solid. To avoid the typical spurious modes of this formulation we introduce a nonconforming discretization. Error estimates for the approximation of eigenvalues and eigenvectors are given.