Numerical Methods for the First Biharmonic Equation and for the Two-Dimensional Stokes Problem

Abstract

This paper is devoted to the numerical solution of the first biharmonic problem $\Delta ^2 \psi = f$, $\psi |_\Gamma = g_1 $, ${{\partial \psi } / {\partial n}}|_\Gamma = g_2 $ as a coupled system of harmonic problems involving the vorticity $\omega = - \Delta \psi $. The main difficulty in this decomposition lies in the fact that the trace $\omega |_\Gamma $ is unknown. A part of the paper is devoted to the analysis of the continuous problem and the relations between $\omega |_\Gamma $ and ${{\partial \psi } / {\partial n}}|_\Gamma $ are studied. By using a suitable mixed finite element approximation, similar properties for the discrete problem are found from which it follows that the trace of the discrete vorticity is the solution of a linear system whose matrix is symmetric, positive definite, but unknown. Then various methods for solving this system are studied. In one of them the matrix of the system is constructed via the solution of approximate Dirichlet problems for $ - \Delta $. Iterative methods are also studied, for which we don’t have to construct the above matrix if we solve, at each iteration, two approximate Dirichlet problems; among these iterative methods we describe a conjugate gradient algorithm which seems new in the context. The approximations and algorithms to be described contain and generalize some well known techniques related to finite difference approximations of the first biharmonic problem. Extension of these various methods to the Stokes problem on two-dimensional p-connected domains (with $p \geqq 1$) is also studied.