Discontinuous-Skeletal methods with linear and quadratic reconstructions for the elliptic obstacle problem

Discontinuous-skeletal methods are introduced and analyzed for the elliptic obstacle problem in two and three space dimensions. The methods are formulated in terms of face unknowns which are polynomials of degree k = 0 or k = 1 and in terms of cell unknowns which are polynomials of degree l = 0. The discrete obstacle constraints are enforced on the cell unknowns. A priori error estimates of optimal order (up to the regularity of the exact solution) are shown. Specifically, for k = 0, the method employs a local linear reconstruction operator and achieves an energy-error estimate of order h, where h is the mesh-size, whereas for k = 1, the method employs a local quadratic reconstruction operator and achieves an energy-error estimate of order $h^{3/2 − \epsilon}$, $\epsilon > 0$. Numerical experiments in two and three space dimensions illustrate the theoretical results.