We present a multi-grid method for a class of structured generalized Lyapunov matrix equations. Such equations need to be solved in each step of the Newton method for algebraic Riccati equations, which arise from linear-quadratic optimal control problems governed by partial differential equations. We prove the rate of convergence of the two-grid method to be bounded independent of the dimension of the problem under certain assumptions. The multi-grid method is based on matrix-matrix multiplications and thus it offers a great potential for a parallelization. The efficiency of the method is demonstrated by numerical experiments.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / The subject of elastohydrodynamic lubrication is identified with situations in which elastic deformation plays a significant role in the hydrodynamic lubrication process. To analyze the elastohydrodynamic lubrication (LEH) of elliptical contacts it is necessary to solve simultaneously the Reynolds equation for pressure and the elasticity equation. The greatest shortcoming of this approach is the dependence exponential between the viscosity and the pressure which is, so that the coupled equations system becomes highly nonlinear. The objective of this work is to analyze LEH in elliptical contacts by means of multi grid algorithms (used for modeling and simulating complex dynamic systems) to solve them. / O estudo da Lubrificação Elasto Hidro Dinâmica (LEH) está relacionado às situações onde a deformação elástica dos corpos em contato hidrodinâmico não pode ser desprezada. Para analisar a LEH em contatos elípticos é necessária a solução simultânea da equação de Reynolds para a pressão e da equação da elasticidade. Para a deformação complementa-se esta formulação com a adoção de uma relação exponencial da viscosidade com a pressão, o que torna o sistema de equações acopladas altamente não-linear. O objetivo deste trabalho é o de analisar a LEH em contatos elípticos, resolvendo-a com o auxilio da técnica de Multi Grelha, que é um acelerador de convergência dos sistemas lineares gerados na discretizacão das equações que regem o fenômeno. / Mestre em Engenharia Mecânica

In the paper, the parallelization of multi-grid methods for solving second-order elliptic boundary value problems in two-dimensional domains is discussed. The parallelization strategy is based on a non-overlapping domain decomposition data structure such that the algorithm is well-suited for an implementation on a parallel machine with MIMD architecture. For getting an algorithm with a good paral- lel performance it is necessary to have as few communication as possible between the processors. In our implementation, communication is only needed within the smoothing procedures and the coarse-grid solver. The interpolation and restriction procedures can be performed without any communication. New variants of smoothers of Gauss-Seidel type having the same communication cost as Jacobi smoothers are presented. For solving the coarse-grid systems iterative methods are proposed that are applied to the corresponding Schur complement system. Three numerical examples, namely a Poisson equation, a magnetic field problem, and a plane linear elasticity problem, demonstrate the efficiency of the parallel multi- grid algorithm.