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A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems. (English) Zbl 1114.74058

Summary: Nonconforming domain decomposition methods and their application to the numerical simulation of nonlinear multibody contact problems play an important role in many applications in mechanics. To handle the nonlinearity of the contact conditions, we apply a primal-dual active set strategy based on dual Lagrange multipliers. Combining this method with an optimal multigrid for the resulting linear algebraic problems and using inexact strategies, our algorithm yields an efficient iterative solver. Furthermore, we establish, under some regularity assumptions on the solution, optimal convergence orders for the discretization errors for the displacement and the Lagrange multiplier for linear and quadratic finite element spaces; we combine quadratic finite elements with linear and quadratic dual Lagrange multipliers. Several numerical examples confirm our theoretical results. In the last section, we extend our algorithm to a dynamic nonlinear multibody contact problem.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
70E55 Dynamics of multibody systems
74M15 Contact in solid mechanics

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