  The conditioning of implicit Runge-Kutta (RK) integration for linear finite
element approximation of diffusion equations on general anisotropic meshes is
investigated. Bounds are established for the condition number of the resulting
linear system with and without diagonal preconditioning for the implicit Euler
and general implicit RK methods. Two solution strategies are considered for the
linear system resulting from general implicit RK integration: the simultaneous
solution (the system is solved as a whole) and a successive solution which
follows the commonly used implementation of implicit RK methods to first
transform the system into smaller systems using the Jordan normal form of the
RK matrix and then solve them successively.
  For the simultaneous solution in case of a positive semidefinite symmetric
part of the RK coefficient matrix and for the successive solution it is shown
that . If the smallest eigenvalue of the symmetric part of the RK coefficient
matrix is negative and the simultaneous solution strategy is used, an upper
bound on the time step is given so that the system matrix is positive definite.
  The obtained bounds for the condition number have explicit geometric
interpretations and take the interplay between the diffusion matrix and the
mesh geometry into full consideration. They show that there are three
mesh-dependent factors that can affect the conditioning: the number of
elements, the mesh nonuniformity measured in the Euclidean metric, and the mesh
nonuniformity with respect to the inverse of the diffusion matrix. They also
reveal that the preconditioning using the diagonal of the system matrix, the
mass matrix, or the lumped mass matrix can effectively eliminate the effects of
the mesh nonuniformity measured in the Euclidean metric. Numerical examples are
given.
