Oleg Davydov

Oleg Davydov, Robert Schaback

This paper proves that the approximation of pointwise derivatives of order $s$ of functions in Sobolev space $W_2^m(\R^d)$ by linear combinations of function values cannot have a convergence rate better than $m-s-d/2$, no matter how many nodes are used for approximation and where they are placed. These convergence rates are attained by {\em scalable} approximations that are exact on polynomials of order at least $\lfloor m-d/2\rfloor +1$, proving that the rates are optimal for given $m,\,s,$ and $d$. And, for a fixed node set $X\subset\R^d$, the convergence rate in any Sobolev space $W_2^m(Ω)$ cannot be better than $q-s$ where $q$ is the maximal possible order of polynomial exactness of approximations based on $X$, no matter how large $m$ is. In particular,scalable stencil constructions via polyharmonic kernels are shown to realize the optimal convergence rates, and good approximations of their error in Sobolev space can be calculated via their error in Beppo-Levi spaces. This allows to construct near-optimal stencils in Sobolev spaces stably and efficiently, for use in meshless methods to solve partial differential equations via generalized finite differences (RBF-FD). Numerical examples are included for illustration.

Oleg Davydov, Robert Schaback

We investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector. Error bounds are given in terms of a growth function that carries the information about the geometry of the centers. Specific forms of weighted $\ell_1$ and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments.

Oleg Davydov, Abid Saeed

We introduce bivariate $C^1$ piecewise quintic finite element spaces for curved domains enclosed by piecewise conics satisfying homogeneous boundary conditions, construct local bases for them using Bernstein-Bézier techniques, and demonstrate the effectiveness of these finite elements for the numerical solution of the Monge-Ampère equation over curved domains by Böhmer's method.

Oleg Davydov, Wee Ping Yeo

We develop a Hermite interpolation scheme and prove error bounds for $C^1$ bivariate piecewise polynomial spaces of Argyris type vanishing on the boundary of curved domains enclosed by piecewise conics.

Cesare Bracco, Oleg Davydov, Carlotta Giannelli et al.

We present a method to detect discontinuity curves, usually called faults, from a set of scattered data. The scheme first extracts from the data set a subset of points close to the faults. This selection is based on an indicator obtained by using numerical differentiation formulas with irregular centers for gradient approximation, since they can be directly applied to the scattered point cloud without intermediate approximations on a grid. The shape of the faults is reconstructed through local computations of regression lines and quadratic least squares approximations. In the final reconstruction stage, a suitable curve interpolation algorithm is applied to the selected set of ordered points previously associated with each fault.

Dang Thi Oanh, Oleg Davydov, Hoang Xuan Phu

We describe and test numerically an adaptive meshless generalized finite difference method based on radial basis functions that competes well with the finite element method on standard benchmark problems with reentrant corners of the boundary, sharp peaks and rapid oscillations in the neighborhood of an isolated point. This is achieved thanks to significant improvements introduced into the earlier algorithms of [Oleg Davydov and Dang~Thi Oanh, Adaptive meshless centers and RBF stencils for Poisson equation, Journal of Computational Physics, 230:287--304, 2011], including a new error indicator of Zienkiewicz-Zhu type.

Oleg Davydov, Georgii Kostin, Abid Saeed

We consider bivariate piecewise polynomial finite element spaces for curved domains bounded by piecewise conics satisfying homogeneous boundary conditions, construct stable local bases for them using Bernstein-Bézier techniques, prove error bounds and develop optimal assembly algorithms for the finite element system matrices. Numerical experiments confirm the effectiveness of the method.