Approximate solution of the Signorini problem by the finite element method in three-dimensional space

The finite element method is usually used for two-dimensional space. The paper investigates the finite element method for solving the Signorini problem in three-dimensional space.

Free Boundary Regularity for Almost Every Solution to the Signorini Problem

We investigate the regularity of the free boundary for the Signorini problem in $${\mathbb {R}}^{n+1}$$ R n + 1 . It is known that regular points are $$(n-1)$$ ( n - 1 ) -dimensional and $$C^\infty $$ C ∞ . However, even for $$C^\infty $$ C ∞ obstacles $$\varphi $$ φ , the set of non-regular (or degenerate) points could be very large—e.g. with infinite $${\mathcal {H}}^{n-1}$$ H n - 1 measure. The only two assumptions under which a nice structure result for degenerate points has been established are when $$\varphi $$ φ is analytic, and when $$\Delta \varphi < 0$$ Δ φ < 0 . However, even in these cases, the set of degenerate points is in general $$(n-1)$$ ( n - 1 ) -dimensional—as large as the set of regular points. In this work, we show for the first time that, “usually”, the set of degenerate points is small. Namely, we prove that, given any $$C^\infty $$ C ∞ obstacle, for almost every solution the non-regular part of the free boundary is at most $$(n-2)$$ ( n - 2 ) -dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian $$(-\Delta )^s$$ ( - Δ ) s , and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is $$(n-1-\alpha _\circ )$$ ( n - 1 - α ∘ ) -dimensional for almost all times t, for some $$\alpha _\circ > 0$$ α ∘ > 0 . Finally, we construct some new examples of free boundaries with degenerate points.

Convergence Analysis of a Discontinuous Finite Volume Method for the Signorini Problem

We introduce and analyze a discontinuous finite volume method for the Signorini problem. Under suitable regularity assumptions on the exact solution, we derive an optimal a priori error estimate in the energy norm.