Direct discretizations with applications to meshless methods for PDEs

A central problem of numerical analysis is the approximate evaluation of integrals or derivatives of functions. In more generality, this is the approximate evaluation of a linear functional defined on a space of functions. Users often just have values of a function u at scattered points x1, . . . ,xN in the domain Ω of u, and then the value λ (u) of a linear functional λ must be approximated via direct approximation formulae λ (u)≈ N ∑ j=1 a ju(x j), i.e. we approximate λ by point evaluation functionals δx j : u 7→ u(x j). Such direct discretizations include classical cases like Newton–Cotes integration formulas or divided differences as approximations of derivatives. They are central for many methods solving partial differential equations, and their error analysis has a long– standing history going back to Peano and his kernel theorem. They also have a strong connection to Approximation Theory. Here, we apply certain optimizations to certain classes of such discretizations, and we evaluate error norms in Beppo–Levi– and Sobolev spaces. This allows to compare discretizations of very different types. including those that are based on exactness on polynomials and those which are by definition optimal on certain function spaces but lack sparsity. Special attention is given to discretizations that are used within current meshless methods for solving partial differential equations. Much of this work is based on recent collaboration with Oleg Davydov of the University of Strathclyde, Scotland, and Davoud Mirzaei of the University of Isfahan, Iran. 2000 AMS subject classification: 65M06, 65N06, 65D15, 65D25, 65D30, 65D32, 65D0, 41A05, 41A10, 41A55, 42A82 .