Convergence of the discrete free boundaries for finite element approximations. (English) Zbl 0547.65081

Brezzi, F.; Caffarelli, L. A.

RAIRO, Anal. Numér. 17, 385-395 (1983).

The authors summarize their results in the following manner. ”We show, on a model ’obstacle problem’ that the discrete (piecewise linear) finite element free boundary converges to the free boundary of the continuous problem with a rate which is approximately the square root of the \(L^{\infty}\) distance between the continuous and the discrete solution.” The model problem involves the Laplacian and the discretization uses simplices with the regularity property that the projection of a given vertex P onto its opposing hyperplane falls in the closure of the opposing face. The obstacle is zero. Under these conditions, a discrete maximum principle holds. An important consequence of the maximum principle is an estimate from both above and below of the discrete Laplacian of the square of the distance function \(\sigma_ Q(x)=| x-Q|^ 2\), where Q is some point. The second author has shown [Bull. Unione Mat. Ital., V. Ser., A 18, 109-113 (1981; Zbl 0453.35085)] that the solution to the continuous problem leaves the obstacle at a minimum rate. The estimate mentioned above allows the authors to prove a similar result for the solution of the discrete problem. Upon assuming that an \(L^{\infty}\) error estimate is known for the difference between the continuous and discrete solutions, the authors are able to deduce estimates of the rate of convergence of the approximate free boundary to the true one.

Reviewer: M.Sussman

MSC:

65Z05	Applications to the sciences
65K10	Numerical optimization and variational techniques
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35R35	Free boundary problems for PDEs
49J40	Variational inequalities

Keywords:

obstacle problem; finite element; free boundary; Laplacian; discrete maximum principle; rate of convergence