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On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations. (English) Zbl 0645.65066
The 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.
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
Software:
symrcm
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References:
[1] Gantmacher, F., The theory of matrices, (1960), Chelsea, New York · JFM 65.1131.03
[2] Golub, G.; Mayers, D., Use of preconditioning over irregular regions, (), 3-14
[3] Gunzburger, M.; Nicolaides, R., Elimination with noninvertible pivots, Linear algebra appl., 64, 183-189, (1985) · Zbl 0599.65013
[4] Girault, V.; Raviart, P.-A., Finite element approximation of the navier—stokes equations, (1979), Springer Berlin · Zbl 0413.65081
[5] Boland, J.; Nicolaides, R., Stability of finite elements under divergence constraints, SIAM J. numer. anal., 20, 722-731, (1983) · Zbl 0521.76027
[6] Boland, J.; Nicolaides, R., Stable and semistable low order finite elements for viscous flows, SIAM J. numer. anal., 22, 474-492, (1985) · Zbl 0578.65123
[7] Fix, G.; Gunzburger, M.; Nicolaides, R., On mixed finite element methods for first-order elliptic systems, Numer. math., 37, 29-48, (1981) · Zbl 0459.65072
[8] Adams, L.; Voigt, R., A methodology for exploiting parallelism in the finite element process, (), 373-392
[9] George, A.; Liu, J., Computer solution of large sparse positive definite systems, (1981), Prentice Hall Englewood Cliffs, NJ · Zbl 0516.65010
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