On the linear convergence of additive Schwarz methods for the p-Laplacian
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We consider additive Schwarz methods for boundary value problems involving the p-Laplacian. While the existing theoretical estimates for the convergence rate of the additive Schwarz methods for the p-Laplacian are sublinear, the actual convergence rate observed by numerical experiments is linear. In this paper, we close the gap between these theoretical and numerical results; we prove the linear convergence of the additive Schwarz methods for the p-Laplacian. The linear convergence of the methods is derived based on a new convergence theory written in terms of a distance-like function that behaves like the Bregman distance of the convex energy functional associated to the problem. The result is then further extended to handle variational inequalities involving the p-Laplacian as well.
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