# zbMATH — the first resource for mathematics

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 operators 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 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
Full Text:
##### References:
 [1] Dmitrieva, M.V.; Ivanov, A.A.; Tishkin, V.F.; Favorskii, A.P., Construction and investigation of support-operators finite-difference schemes for Maxwell equations in cylindrical geometry, (1985), Keldysh Inst. of Appl. Math. the USSR Ac. of Sc.,, (in Russian) [2] Favorskii, A.P.; Tishkin, V.F.; Shashkov, M.Yu., Variational-difference schemes for the heat conduction equation on non-regular grids, Soviet. phys. dokl., 24, 446-448, (1979) · Zbl 0435.65081 [3] Favorskii, A.P.; Korshiya, T.K.; Tishkin, V.F.; Shashkov, M.Yu., Difference schemes for equations of electro-magnetic field diffusion with anisotropic conductivity coefficients, (1980), Keldysh Inst. of Appl. Math. the USSR Ac. of Sc.,, (in Russian) · Zbl 0473.65068 [4] Favorskii, A.P.; Korshiya, T.K.; Shashkov, M.Yu.; Tishkin, V.F., Variational approach to the construction of finite-difference schemes for the diffusion equations for magnetic field, Differential equations, 18, 7, 863-872, (1982) [5] Favorskii, A.P.; Korshiya, T.K.; Shashkov, M.Yu.; Tishkin, V.F., A variational approach to the construction of difference schemes on curvilinear meshes for heat-conduction equation, Comput. math. math. phys., 20, 135-155, (1980) · Zbl 0473.65068 [6] Hyman, J.M.; Shashkov, M.Yu., Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, Comput. math. appl., 33, 4, 81-104, (1997) · Zbl 0868.65006 [7] J.M. Hyman and M.Yu. Shashkov, The orthogonal decomposition theorems for mimetic finite difference methods, Report LA-UR-96-4735 of Los Alamos National Laboratory, Los Alamos, NM; also: SIAM J. Numer. Anal., submitted. [8] Hyman, J.M.; Shashkov, M.Yu.; Steinberg, S., The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials, J. comput. phys., 132, 130-148, (1997) · Zbl 0881.65093 [9] Hyman, J.M.; Shashkov, M.Yu.; Steinberg, S., Problems with heterogeneous and non-isotropic media or distorted grids, (), 249-260 [10] Knupp, P.M.; Steinberg, S., The fundamentals of grid generation, (1993), CRC Press, Boca Raton, FL [11] Lele, S.K., Compact finite difference schemes with spectral-like resolution, J. comput. phys., 103, 16-42, (1992) · Zbl 0759.65006 [12] Leventhal, S.H., An operator compact implicit method of exponential type, J. comput. phys., 46, 138-165, (1982) · Zbl 0514.76086 [13] Lynch, R.E.; Rice, J.R., A high-order difference method for differential equations, Math. comp., 34, 333-372, (1980) · Zbl 0424.65037 [14] Samarskii, A.A.; Tishkin, V.F.; Favorskii, A.P.; Shashkov, M.Yu., Employment of the reference-operator method in the construction of finite-difference analogs of tensor operations, Differential equations, 18, 881-885, (1982) · Zbl 0532.65069 [15] Samarskii, A.A.; Tishkin, V.F.; Favorskii, A.P.; Shashkov, M.Yu., Operational finite-difference schemes, Differential equations, 17, 854-862, (1981) · Zbl 0485.65060 [16] Shashkov, M.Yu., Conservative finite-difference schemes on general grids, (1995), CRC Press, Boca Raton, FL [17] Shashkov, M.Yu.; Steinberg, S., Solving diffusion equations with rough coefficients in rough grids, J. comput. phys., 129, 383-405, (1996) · Zbl 0874.65062 [18] Shashkov, M.Yu.; Steinberg, S., Support-operator finite-difference algorithms for general elliptic problems, J. comput. phys., 118, 131-151, (1995) · Zbl 0824.65101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.