Spline approximation of explicit surfaces containing irregularities

We consider the problem of approximation of a non regular a functions, i.e. functions which are discontinuous or which have some discontinuous derivatives on a subset F of an open, bounded set Omega in R-n, and that belong to the Sobolev space H-m(Omega\(F) over bar) for some integer m > n/2. The question is to construct, from Lagrange of 1(st) order Hermite data of a non regular function f, an approximant of f of class C-k on Omega\(F) over bar with k = 1 or 2. The standard example of this situation is the modelling of geological surfaces (cf J. Springer [24]). The answer we provide to the problem of approximation of non regular functions is obtained by adapting the theory of D-m-splines over a bounded open set in R-n. We first define the D-m-splines over Omega' = Omega\(F) over bar and then, introducing a suitable finite element space, the ''discrete D-m-splines over Omega''' : these are the functions we propose for the approximation of non regular functions. Finally, we study the convergence of the discrete smoothing D-m-splines over Omega' and we give some numerical results.