A full multi-grid method for the solution of the cell vertex finite volume Cauchy--Riemann equations A. Borz\`{\i}, K.W. Morton, E. S\"uli and M. Vanmaele Oxford University Computing Laboratory Numerical Analysis Group Wolfson Building, Parks Road Oxford, England OX1 3QD Abstract The system of inhomogeneous Cauchy--Riemann equations defined on a square domain and subject to Dirichlet boundary conditions is considered. This problem is discretised by using the cell vertex finite volume method on quadrilateral or triangular meshes. The resulting algebraic problem is overdetermined and the solution is defined in a least squares sense. By this approach a consistent algebraic problem is obtained which differs from the original one by $O(h^2)$ perturbations of the right-hand side. A suitable cell-based convergent smoothing iteration is defined which is naturally linked to the least squares formulation. Hence a standard multi-grid algorithm is presented and discussed which combines the given smoother and cell-based transfer operators. Some remarkable reduction properties of these operators are examined. A full multi-grid method is constructed which solves the discrete problem to the level of truncation error by employing one multi-grid cycle at each current level of discretisation. Experiments and applications of the full multi-grid scheme are presented.