A multigrid optimization approach for the numerical solution of a class of variational inequalities of the second kind

Abstract

In this thesis we introduce a Multigrid Optimization Algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind, which involve the p-Laplacian operator and the L1-norm of the gradient. This approach follows from the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional, Therefore, we proposed a Huber regularization of the functional and a finite element discretization for the problem. Further, we analyze the regularity of the discretized energy functional, and we are able to prove that its Jacobian is slantly differentiable. This regularity property is useful to analyze the convergence of the MG/OPT algorithm. In fact, we demostrate that the algorithm is global convergent by using the mean value theorem for slantly differentiable functions. Finally, we analyze the performance of the MG/OPT algorithm when used to simulate the visco-plastic flow of Bingham, Casson and Herschel-Bulkley fluids in a pipe. Several numerical experiments are carried out to show the main features of the proposed method.