In this paper, a multigrid algorithm is developed for the third-order accurate solution of Cauchy–Riemann equations discretized in the cell-vertex finite-volume fashion: the solution values stored at vertices and the residuals defined on triangular elements. On triangular grids, this results in a highly overdetermined problem, and therefore we consider its solution that minimizes the residuals in the least-squares norm. The standard second-order least-squares scheme is extended to third-order by adding a high-order correction term in the residual. The resulting high-order method is shown to give sufficiently accurate solutions on relatively coarse grids. Combined with a multigrid technique, the method then becomes a highly accurate and efficient solver. We present some results to demonstrate its accuracy and efficiency, including both structured and unstructured triangular grids. Copyright © 2006 John Wiley & Sons, Ltd.