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.