On the convergence of Uzawa's method for the solution of biharmonic problem (Q760769)

The author considers the numerical solution of the biharmonic problem with linear and quadratic mixed finite elements. The approximate problems are solved by Uzawa's method. Explicit asymptotic convergence rates are shown and from these a lower bound on the number n of iterations is obtained that are needed to have a truncation error for the iteration of the same order as the discretization error. The bound turns out as \(n\geq ch^{-1}| \ln h|\) for linear finite elements and roughly four times that for quadratic elements. It is, however, assumed that the parameter in Uzawa's method is chosen equal to an in general unknown quantity for which only a lower bound is quoted from the literature.