The solutions of the Poisson equation in regular and irregular shaped physical domains are obtained by the cubature method. The solutions of the three test problems involving regular shaped domains are compared with the analytical solutions and the control-volume, five-point finite difference, Galerkin finite element and quadrature numerical solutions. The application of the cubature method for an irregular shaped domain is illustrated by solving the pool boiling in a cylindrical cavity, and the solutions are compared by the quadrature solution. It is shown in all cases that the cubature method can generate significantly more accurate results in a convenient manner with much less computational effort. © 1997 John Wiley & Sons, Ltd.