The generalized MSR-method (GMSR-method) is proposed as a finite element fluid analysis algorithm for arbitrarily deformed elements using the error analysis approach. The MSR-method was originally developed by one of the authors in our previous research works using a modified Galerkin method (MGM) for a convection–diffusion equation and the SIMPLER-approach. In this paper, this MGM is developed theoretically in the case of arbitrarily deformed elements using the error analysis approach. In the GMSR-method, since the inertia term and the pressure term are considered explicitly, only symmetrical matrices appear. Hence, it helps us reduce computational memory and computation time. Moreover, artificial viscosity and diffusivity are introduced through an error analysis approach to improve the accuracy and stability. This GMSR-method is applied for two- and three-dimensional natural convection problems in a cavity. In the computations at different Rayleigh numbers, it is shown that this method gives reasonable results compared to other research works. Thus, it is found that the GMSR-method is applicable to thermal-fluid flow problems with complicated boundaries. Copyright © 2004 John Wiley & Sons, Ltd.