Residual Error Estimators for Coulomb Friction

Abstract

This paper is concerned with residual error estimators for finite element approximations of Coulomb frictional contact problems. A recent uniqueness result by Renard in [72] for the continuous problem allows us to perform an a posteriori error analysis. We propose, study and implement numerically two residual error estimators associated with two finite element discretizations. In both cases the estimators permit to obtain upper and lower bounds of the discretization error.

Onhp-adaptive BEM for frictional contact problems in linear elasticity

“…Here, ∥v∥ 2 W := ⟨W v, v⟩ and ∥ψ∥ 2 V := ⟨V ψ, ψ⟩ are the involved energy norms and u hp+1 ∈ V hp+1 is the Galerkin approximation of (30). In order to avoid the computation of a global problem and to localize the error a two-level additive Schwarz operator is used.…”

“…Let (u, λ), (u hp , λ hp ) and z solve (8), (12), (30) with the Coulomb friction modified Lagrange multiplier sets (56) and (57). Under the assumption that ∥F ∥ L ∞ (Γ C ) is sufficiently small there exists a constantC > 0 independent of h and p such…”

“…Let (u, λ), (u hp , λ hp ) and z solve (8),(12),(30), respectively. Then there exists a constant C > 0 independent of h and p such that…”

“…(see [30]), F ≥ 0 constant and F ∥ξ ∥ sufficiently small, then there exists a constantC > 0 independent of h and p such that…”

“…Compared to the Tresca frictional case, only the part involving the tangential part of the Lagrange multiplier must be changed which can be handled analogously to [30].…”