Analysis of a new mixed formulation of the obstacle problem

Abstract:

For the obstacle problem:

for which the unknowns are u and the free boundary ∂[u > ψ]. It is known¹ that the coincidence set [u = ψ] is contained in the set [f - ∆ψ ≥ 0]. In this work we introduce µ = (f - ∆ψ)⁺_χ[u>ψ], that characterizes the domain of non contact,² and we analyze a mixed formulation of the obstacle problem where µ appears as a Lagrange multiplier. We show, in particular, that the solution µ_h of the approached (by linear finite element) mixed problem converges to µ with an order of convergence that is an O(h), which is optimal seen the equivalent result on the approximation of the free boundary³ and.⁴