Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem

TLDR: In this paper, the first bi-harmonic problem on general two-dimensional domains was solved using a mixed finite element method, where the continuous problem has been approximated by an appropriate mixed-finite element method.

Abstract: We describe in this report various methods, iterative and "almost direct," for solving the first biharmonic problem on general two-dimensional domains once the continuous problem has been approximated by an appropriate mixed finite element method. Using the approach described in this report we recover some well known methods for solving the first biharmonic equation as a system of coupled harmonic equations, but some of the methods discussed here are completely new, including a conjugate gradient type algorithm. In the last part of this report we discuss the extension of the above methods to the numerical solution of the two dimensional Stokes problem in p- connected domains (p $\geq$ 1) through the stream function-vorticity formulation.