On the $O(1/n)$ Convergence Rate of the Douglas-Rachford Alternating Direction Method
SIAM Journal on Numerical Analysis
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We describe in this report the numerical analysis of a particular class of nonlinear Dirichlet problems. We consider an equivalent variational inequality formulation on which the problems of existence, uniqueness and approximation are easier to discuss. We prove in particular the convergence of an approximation by piecewise linear finite elements. Finally, we describe and compare several iterative methods for solving the approximate problems and particularly some new algorithms of augmented lagrangian type, which contain as special case some well-known alternating direction methods. Numerical results are presented.