The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials.

*(English)*Zbl 0881.65093The paper is the third of a series on the support-operators method [see M. Shashkov and S. Steinberg, J. Comput. Phys. 118, No. 1, 131-151 (1995; Zbl 0824.65101) and ibid. 129, No. 2, 383-40, (1996; Zbl 0874.65062)].

The authors describe and investigate a new second-order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non-isotropic media.

At first, the boundary value problem is formulated as a system of first-order equations. This formulation is given in operator form to illuminate the properties of the operators that should have analogs in the discrete case. Then, the construction of logically rectangular grids and the discretization of scalar and vector functions are described. Here, cell-centered discretizations of scalar functions, and both nodal and face-centered discretizations of vector functions (e.g. the flux) are used. Following the support-operators method, approximations for \(\text{div}\) and \(K \text{grad}\) (\(K\) is the conductivity matrix) are derived in the cases of the nodal and surface discretizations. Based on these discrete operators the finite difference scheme for \(\text{div} K \text{grad}\) is constructed.

Furthermore, iterative methods for solving the discrete problems are discussed.

Finally, the presented algorithms are compared on five examples. The experiments show that the surface discretization approach performs reliably on all examples, and the nodal discretization gives reasonable results.

The authors describe and investigate a new second-order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non-isotropic media.

At first, the boundary value problem is formulated as a system of first-order equations. This formulation is given in operator form to illuminate the properties of the operators that should have analogs in the discrete case. Then, the construction of logically rectangular grids and the discretization of scalar and vector functions are described. Here, cell-centered discretizations of scalar functions, and both nodal and face-centered discretizations of vector functions (e.g. the flux) are used. Following the support-operators method, approximations for \(\text{div}\) and \(K \text{grad}\) (\(K\) is the conductivity matrix) are derived in the cases of the nodal and surface discretizations. Based on these discrete operators the finite difference scheme for \(\text{div} K \text{grad}\) is constructed.

Furthermore, iterative methods for solving the discrete problems are discussed.

Finally, the presented algorithms are compared on five examples. The experiments show that the surface discretization approach performs reliably on all examples, and the nodal discretization gives reasonable results.

Reviewer: M.Jung (Chemnitz)

##### MSC:

65N06 | Finite difference methods for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

35J25 | Boundary value problems for second-order elliptic equations |

##### Keywords:

diffusion equation; finite difference discretization; support-operators method; iterative solvers
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\textit{J. Hyman} et al., J. Comput. Phys. 132, No. 1, 130--148 (1997; Zbl 0881.65093)

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