Application of the adjoint operators in solving the Divergence-Curl problems

Abstract

We create a discrete analog of vector analysis on logically rectangular, nonorthogonal, nonsmooth girds by using the support-operator method. Then we can define Natural discrete analog of the divergence, gradient, and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates. But it is impossible to construct discrete analogs of the second-order operators divgrad, graddiv, and curlcurl because of incompatibilities in domains and in the ranges of values for the operators. However, the adjoint operators have complementary domains and ranges of values and the combined set of natural and adjoint operators allow a consistent formulation for all the compound discrete operators. We use this operators efficiently then we can solve many differential problems.