In this paper, we show that the additive Schwarz method proposed in [3] to solve one-obstacle problems converges in a much more general framework. We prove that this method can be applied to the minimization of functionals over a general enough convex set in a reflexive Banach space. In the Sobolev spaces, the proposed method is an additive Schwarz method for the solution of the variational inequalities coming from the minimization of non-quadratic functionals. Also, we show that the one-, two-level variants of the method in the finite element space converge, and we explicitly write the constants in the error estimations depending on the overlapping and mesh parameters.
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Badea, L. (2008). An Additive Schwarz Method for the Constrained Minimization of Functionals in Reflexive Banach Spaces. In: Langer, U., Discacciati, M., Keyes, D.E., Widlund, O.B., Zulehner, W. (eds) Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75199-1_54
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DOI: https://doi.org/10.1007/978-3-540-75199-1_54
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