Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids.(English)Zbl 1005.65024
Summary: We use the support-operator method to derive new discrete approximations of the divergence, gradient, and curl using discrete analogs of the integral identities satisfied by the differential operators. These new discrete operators are adjoint to the previously derived natural discrete operators defined using ‘natural’ coordinate-invariant definitions, such as Gauss’ theorem for the divergence.
The natural operators cannot be combined to construct discrete analogs of the second-order operatorsdiv grad, grad div, andcurl 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 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 also prove that the operators satisfy discrete analogs of the major theorems of vector analysis relating the differential operators, including \(\mathbf{div} {\overset\rightarrow {\mathbf A}}=0\) if and only if \({\overset\rightarrow{\mathbf A}}=\mathbf{curl} {\overset\rightarrow{\mathbf B}}; \mathbf{curl} {\overset\rightarrow{\mathbf A}} =0\) if and only if \({\overset\rightarrow{\mathbf A}}=\mathbf{grad}\varphi\).
The natural operators cannot be combined to construct discrete analogs of the second-order operatorsdiv grad, grad div, andcurl 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 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 also prove that the operators satisfy discrete analogs of the major theorems of vector analysis relating the differential operators, including \(\mathbf{div} {\overset\rightarrow {\mathbf A}}=0\) if and only if \({\overset\rightarrow{\mathbf A}}=\mathbf{curl} {\overset\rightarrow{\mathbf B}}; \mathbf{curl} {\overset\rightarrow{\mathbf A}} =0\) if and only if \({\overset\rightarrow{\mathbf A}}=\mathbf{grad}\varphi\).
MSC:
| 65D25 | Numerical differentiation |
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References:
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