Multilevel Uzawa and Arrow–Hurwicz Algorithms for General Saddle Point Problems

Lori Badea ()
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Lori Badea: Institute of Mathematics of the Romanian Academy

Journal of Optimization Theory and Applications, 2023, vol. 198, issue 2, No 9, 678-709

Abstract: Abstract In this paper, we introduce and analyze multilevel inexact Uzawa and Arrow–Hurwicz algorithms for solving saddle point problems. For the definition of the problem and that of the Uzawa and Arrow–Hurwicz algorithms, we adopt the framework introduced in I. Ekeland and R. Temam, convex analysis and variational problems, North-Holland Publishing Company, Amsterdam, Oxford, 1976, where the saddle point is defined by optimizations on convex sets. The results are obtained for Hilbert spaces, and therefore they can be applied to obtain convergence results for the multigrid methods in finite element spaces. We prove the convergence of the two algorithms, the multilevel Uzawa and the multilevel Arrow–Hurwicz algorithms. Also, we give new convergence proofs for the Uzawa and Arrow–Hurwicz algorithms themselves to better characterize their convergence.

Keywords: Multilevel and multigrid methods; Uzawa and Arrow–Hurwicz algorithms; Saddle point problems; Domain decomposition methods; 65J15; 49M27; 49M27; 65N55 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-023-02231-2

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