Volume 22, pp. 41-70, 2006.

A network programming approach in solving Darcy's equations by mixed finite-element methods

M. Arioli and G. Manzini

Abstract

We use the null space algorithm approach to solve the augmented systems produced by the mixed finite-element approximation of Darcy's laws. Taking into account the properties of the graph representing the triangulation, we adapt the null space technique proposed in [M. Arioli and L. Baldini, A backward error analysis of a null space algorithm in sparse quadratic programming, SIAM J. Matrix Anal. and Applics., 23 (2001), pp. 425–442], where an iterative-direct hybrid method is described. In particular, we use network programming techniques to identify the renumbering of the triangles and the edges, which enables us to compute the null space without floating-point operations. Moreover, we extensively take advantage of the graph properties to build efficient preconditioners for the iterative algorithm. Finally, we present the results of several numerical tests.

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Key words

augmented systems, sparse matrices, mixed finite-element, graph theory

AMS subject classifications

65F05, 65F10, 64F25, 65F50, 65G05

Links to the cited ETNA articles

[6]Vol. 22 (2006), pp. 17-40 M. Arioli, J. Maryška, M. Rozložník, and M. Tůma: Dual variable methods for mixed-hybrid finite element approximation of the potential fluid flow problem in porous media
[37]Vol. 13 (2002), pp. 56-80 Zdeněk Strakoš and Petr Tichý: On error estimation in the conjugate gradient method and why it works in finite precision computations

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