The Arrow-Hurwicz algorithm with alternating iterative parameters (Q1589046)

The Stokes problem in hydrodynamics is a typical source of linear equations \(Lz=F\) with a vanishing (or small) lowest diagonal submatrix of \(L\). The natural partitioning of the underlying real and symmetric \(L\) in its diagonal and off-diagonal respective nonvanishing submatrices \(A\) and \(B\) induces a natural partitioning of the whole problem. In practice, its subsequent iterative solution usually has a two- or three-term form of the Richardson or Uzawa-type algorithm, respectively. An ''optimal'' rate of convergence has been specified in many particular cases in the past. The paper shows that all these algorithms may be represented in a common form of a certain Arrow-Hurwitz ''artificial compressibility'' algorithm after a suitable choice of its variable iteration parameters. Under certain conditions it is shown that the same ''optimal'' rate of convergence may be achieved in this case, in spite of an asymmetry of its matrix formulation. The proof is constructive: Explicit definitions of the optimal parameters are given, derived and discussed.