On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations.

*(English)*Zbl 0645.65066The authors consider the solution of a boundary value problem on a region \(\Omega\) by substructuring. By this they mean disecting \(\Omega\) into disjoint regions \(\Omega_ i\), \(i=1,...,m\), with interfaces \(\Gamma_{ij}\). Letting \(U_ i\) be the solution of a finite-difference on finite-element approximation on \(\Omega_ i\), while \(U_ 0\) is the solution on the boundary and interfaces. Then the problem can be written in the block matrix form \(A_ iU_ i+B_ iU_ 0=F_ i,\) \(i=1,...,m\), \(\Sigma C_ iU_ i+A_ 0U_ 0=F_ 0,\) where \(B_ i\) are ng. The construction of a grid for a given domain involves two main steps; generation of a “rough” initial grid by a simple and fast procedure; the grid is improved according to certain specified criteria taking the initial one as a starting iterate.

Moreover adaptive redistribution of the grid subject to select constraints to improve the accuracy of the numerical solution is possible. The cost functional for the optimization step is using smoothness, orthogonality and approximability criteria. A discrete geometric approach and a formal continuous one are presented to handle grid smoothness and orthogonality. Several cost functionals including also adaptive grading of the mesh are considered and compared to those of J. U. Brackbill and J. S. Saltzman [J. Comput. Phys. 46, 342- 368 (1982; Zbl 0489.76007)].

The optimization problems are solved by the Fletcher-Reeves and Polak- Ribière conjugate gradient methods. The ideas are implemented to generate 2D and 3D grids both for interior and exterior domains to a given profile or surface. Examples are considered and numerical results are given. In some cases the initial grid is randomly generated in order to demonstrate the ability of the procedure to redistribute a bad grid. Nice and interesting figures are given.

Moreover adaptive redistribution of the grid subject to select constraints to improve the accuracy of the numerical solution is possible. The cost functional for the optimization step is using smoothness, orthogonality and approximability criteria. A discrete geometric approach and a formal continuous one are presented to handle grid smoothness and orthogonality. Several cost functionals including also adaptive grading of the mesh are considered and compared to those of J. U. Brackbill and J. S. Saltzman [J. Comput. Phys. 46, 342- 368 (1982; Zbl 0489.76007)].

The optimization problems are solved by the Fletcher-Reeves and Polak- Ribière conjugate gradient methods. The ideas are implemented to generate 2D and 3D grids both for interior and exterior domains to a given profile or surface. Examples are considered and numerical results are given. In some cases the initial grid is randomly generated in order to demonstrate the ability of the procedure to redistribute a bad grid. Nice and interesting figures are given.

Reviewer: V.Arnăutu (Iaşi)

##### MSC:

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65F05 | Direct numerical methods for linear systems and matrix inversion |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

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

35J65 | Nonlinear boundary value problems for linear elliptic equations |

##### Keywords:

nonpositive definite, nonsymmetric matrices; stationary Stokes equations; Helmholtz equation; Poisson equation; singular matrix; block Gaussian elimination; substructuring; cost functional; smoothness; orthogonality; adaptive grading; conjugate gradient methods##### Software:

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\textit{M. D. Gunzburger} and \textit{R. A. Nicolaides}, Appl. Numer. Math. 2, 243--256 (1986; Zbl 0645.65066)

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