The focus of this paper is the numerical study of the fluid dynamics inside an evaporating liquid sessile droplet. The governing equations are solved numerically using the Galerkin/finite element method (G/FEM) for discretization of the spatial domain and an adaptive finite difference method for discretization in the time domain. A mesh generation method needs to be carefully chosen for this problem, preferably the method which continuously gives sufficient mesh resolution, in the region where the numerical solution needs to be resolved, throughout the drop evaporation time. After comparing two mesh generation methods, the elliptic mesh generation method is preferred over the spine mesh method. Successfully, the droplet shape, and velocity, and pressure profiles inside the droplet are solved simultaneously for the case where inertial forces are important (Reynolds number similar to 1) and where the gravitational force is either absent or present. (c) 2006 Elsevier Ltd. All rights reserved.