ADJOINT OPERATORS FOR THE NATURAL DISCRETIZATIONS OF THE DIVERGENCE, GRADIENT AND CURL ON LOGICALLY RECTANGULAR GRIDS

We use the support-operator method to derive new discrete approximatio ns of the divergence, gradient, and curl using discrete analogs of the integral identities satisfied by the differential operators. These ne w discrete operators are adjoint to the previously derived natural dis crete operators defined using 'natural' coordinate-invariant definitio ns, such as Gauss' theorem for the divergence. The natural operators c annot be combined to construct discrete analogs of the second-order op erators div grad, grad div, and curl curl because of incompatibilities in domains and in the ranges of values for the operators. The same is true for the adjoint operators, However, the adjoint operators have c omplementary domains and ranges of values and the combined set of natu ral and adjoint operators allow a consistent formulation for ail the c ompound discrete operators. We also prove that the operators satisfy d iscrete analogs of the major theorems of vector analysis relating the differential operators, including div (A) over right arrow = 0 if and only if (A) over right arrow = curl (B) over right arrow; curl (A) ove r right arrow = 0 if and only if (A) over right arrow = grad phi. (C) 1997 Elsevier Science B.V.