  I formulate a general finite element method (FEM) for self-gravitating
stellar systems. I split the configuration space to finite elements, and
express the potential and density functions over each element in terms of their
nodal values and suitable interpolating functions. General expressions are then
introduced for the Hamiltonian and phase space distribution functions of the
stars that visit a given element. Using the weighted residual form of Poisson's
equation, I derive the Galerkin projection of the perturbed collisionless
Boltzmann equation, and assemble the global evolutionary equations of nodal
distribution functions. The FEM is highly adaptable to all kinds of potential
and density profiles, and it can deal with density clumps and initially
non-axisymmetric systems. I use ring elements of non-uniform widths, choose
linear and quadratic interpolation functions in the radial direction, and apply
the FEM to the stability analysis of the cutout Mestel disc. I also integrate
the forced evolutionary equations and investigate the disturbances of a stable
stellar disc due to the gravitational field of a distant satellite galaxy. The
performance of the FEM and its prospects are discussed.
