Convergence of adaptive FEM for elliptic obstacle problems

Abstract

We treat the convergence of adaptive lowest-order FEM for some elliptic obstacle problem with affine obstacle χ, i.e.

(\nabla u, \nabla (u − v)) ≤ (f, u − v)

for all v ∈ A := {v ∈ H^1_0 (Ω) : v ≥ χ a.e in Ω}. For steering the adaptive mesh-refinement, we use a residual error estimator

\eta_\ell := ( \rho_\ell^2 + osc_\ell^2 )^1/2 that consists of the estimator from [Braess et al., Numer. Math. ´07] and additionally controls the data oscillations. We extend recent ideas from [Cascon et al., Numer. Anal. ´08] for the unrestricted variational problem to overcome the lack of Galerkin orthogonality. The main result states that an appropriately weighted sum

\Delta_\ell := \epsilon_\ell + \gamma \mu_\ell^2

of energy error, edge residuals, and data oscillations satisfies a contraction property

\Delta_{\ell+1} ≤ \kappa \Delta_\ell for all \ell \in \N

within each step of the adaptive loop. Here, \mu_\ell denotes a second error estimator which is equivalent to \eta_\ell and 0 < γ, κ < 1. This result is superior to a prior result from [Braess et al., Numer. Math. ´07] in two ways: First, it is unnecessary to control the decay of the data oscillations explicitly. Second, our analysis avoids the use of discrete local efficiency so that the local mesh-refinement is fairly arbitrary. In addition, we discuss the generalization to the case of inhomogeneous Dirichlet data and non-affine obstacles χ ∈ H^2 (Ω) and obtain similar results.