Error Estimates in the Approximation of a Free Boundary

Abstract

Free boundary problems appear in a great variety of physical problems. See for instance ^1,2,3 for the mathematical setting of many examples. We shall consider here a model case in which the free boundary is the line (or, more generally, the (n−1) dimensional surface) that splits the domain (say, D) where the solution (say, u) is sought, into D = D+ ∪ D⁰ so that u>0 in D⁺ and u ≡ 0 in D⁰. Many free boundary problems can be reduced to this case. If the free boundary is smooth and u is also smooth separately in each subdomain D⁰ and D⁺, then the global regularity of u will depend on the behaviour of u, in D⁺, near the free boundary: for instance, if u behaves like (d(x))^k+α (k integer and nonnegative, 0<α≤1, d(x) = distance of x from the free boundary) then u ∈ C^k+α(D). Assume now that we have got, by means of some discretization process, a sequence of discrete solutions {u_h}_h which converges to u in the L^∞ norm. We set

$$e\left( h \right): = {\left\| {u - {u_h}} \right\|_{{L^\infty }}} \to 0\,for\,h \to 0$$

(1.1)