  We study the stability of explicit Runge-Kutta methods for high order
Lagrangian finite element approximation of linear parabolic equations and
establish bounds on the largest eigenvalue of the system matrix which
determines the largest permissible time step. A bound expressed in terms of the
ratio of the diagonal entries of the stiffness and mass matrices is shown to be
tight within a small factor which depends only on the dimension and the choice
of the reference element and basis functions but is independent of the mesh or
the coefficients of the initial-boundary value problem under consideration.
Another bound, which is less tight and expressed in terms of mesh geometry,
depends only on the number of mesh elements and the alignment of the mesh with
the diffusion matrix. The results provide an insight into how the interplay
between the mesh geometry and the diffusion matrix affects the stability of
explicit integration schemes when applied to a high order finite element
approximation of linear parabolic equations on general nonuniform meshes.
