On the linear convergence of additive Schwarz methods for the $p$-Laplacian

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.