  A local approach to the time integration of PDEs by exponential methods is
proposed, motivated by theoretical estimates by A.Iserles on the decay of
off-diagonal terms in the exponentials of sparse matrices. An overlapping
domain decomposition technique is outlined, that allows to replace the
computation of a global exponential matrix by a number of independent and
easily parallelizable local problems. Advantages and potential problems of the
proposed technique are discussed. Numerical experiments on simple, yet relevant
model problems show that the resulting method allows to increase computational
efficiency with respect to standard implementations of exponential methods.
