Data-dependent triangulations for scattered data interpolation and finite element approximation

In this paper we review methods for improving the accuracy of approximation by piecewise linear polynomials defined over triangulations to bivariate functions, by adapting the triangulation to the structure of the approximated function. Many criteria for optimizing triangulations as well as efficient algorithms for computing data-dependent triangulations that are locally optimal with respect to the criteria are presented. It is demonstrated that data-dependent triangulations can improve significantly the quality of approximation and that long and thin triangles, which are traditionally avoided, are sometimes necessary to have. Our results are concerned with two processes of approximation: (i) scattered data interpolation to bivariate functions, (ii) finite element approximations to solutions of second-order elliptic partial differential equations. Extensions of these ideas to higher-order approximation schemes as well as extensions to multivariate approximation problems are also discussed.