Keywords: FETI, conjugate gradients, steepest descent.

For the minimization of the eliminated system the conjugate gradient method (CG) is usually used due to its fast convergence property. The convergence property of the CG method is countervailed by its sensitivity to the computation error. FETI-1 therefore uses the direct solver (e.g. the LU decomposition) for the elimination of the primal part of the system. If the subdomains are large, the LU decomposition become both memory and time consuming.

In [1] a new gradient minimization algorithm was introduced. It is very robust yet still fast enough to be practically useful. Its main idea is to use a gradient as a descent direction and to compute the step length using a stochastic distribution over an approximated spectrum of the system.

The CG method in the outer loop of the FETI-1 can be replaced by this new minimization algorithm. If we do so, we are free to use an iterative solver (e.g. the CG method) for the inner loop instead of the LU decomposition. This approach uses much less memory and allows us to control the precision of elimination at each step. In the last section of the article we estimate the influence of the computation error, propose two strategies for the precision control and, finally, present some computation results.

1

L. Pronzato, A. Zhigljavsky, "Gradient algorithms for quadratic optimization with fast convergence rates", Computational Optimization and Applications, 2010. doi:10.1007/s10589-010-9319-5

2

O. Rheinbach, "Parallel iterative substructuring in structural mechanics", Archives of Computational Methods in Engineering, 16(4), 425, 2009. doi:10.1007/s11831-009-9035-4

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