Multilevel Solution of Cell Vertex Cauchy--Riemann Equations

Abstract

In this paper a multilevel algorithm for the solution of the cell vertex finite volume Cauchy--Riemann equations is developed. These equations provide a linear algebraic system obtained by the finite volume cell vertex discretization of the inhomogeneous Cauchy--Riemann equations. Both square and triangular cells are employed. The system of linear equations resulting from the cell vertex discretization is overdetermined and its solution is considered in the least squares sense. By this approach a consistent algebraic problem is obtained which differs from the original one by O(h2) perturbation of the right-hand side. A suitable cell-based convergent smoothing iteration is presented which is naturally linked to the least squares formulation. Hence a standard multilevel scheme is presented and discussed which combines the given smoother and a cell-based transfer operator of the residuals and a node-based prolongation operator of the unknown variables. Some remarkable reduction properties of these operators are shown. A full multilevel algorithm is constructed which solves the discrete problem to the level of truncation error by employing one multilevel cycle at each current level of discretization.