Mimetic methods on measure chains

We introduce the divergence and the gradient for functions defined on a mea sure chain, and this includes as special cases both continuous derivatives and discrete forward differences. It is shown that in one dimension, subjec t to Dirichlet boundary conditions, the divergence and the gradient are neg ative adjoints of each other and that the divergence of the gradient is neg ative semidefinite. These are well-known results in the continuous theory, and hence, mimic those properties also for the case of a general measure ch ain. (C) 2001 Elsevier Science Ltd. All rights reserved.