An Adaptive Least-Squares FEM for Linear Elasticity with Optimal Convergence Rates

Abstract

Adaptive mesh-refining is of particular importance in computational mechanics and established here for the lowest-order locking-free least-squares finite element scheme which solely employs conforming $P_1$ approximations for the displacement and lowest-order Raviart--Thomas approximations for the stress variables. This forms a competitive discretization in particular in three-dimensional linear elasticity with traction boundary conditions although the stress approximation does not satisfy the symmetry condition exactly. The paper introduces an adaptive mesh-refining algorithm based on separate marking and exact solve with some novel explicit a posteriori error estimator and proves optimal convergence rates. The point is robustness in the sense that the crucial input parameters $\Theta$ for the Dorfler marking and $\kappa$ for the separate marking as well as the equivalence constants in the asymptotic convergence rates do not degenerate as the Lame parameter $\lambda$ tends to $\infty$.