Related papers: Average Interpolation Under the Maximum Angle Cond…
We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress…
The fast Ewald methods are widely used to compute the point-charge electrostatic interactions in molecular simulations. The key step that introduces errors in the computation is the particle-mesh interpolation. In this work, the optimal…
Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise…
We investigate the error of periodic interpolation, when sampling a function on an arbitrary pattern on the torus. We generalize the periodic Strang-Fix conditions to an anisotropic setting and provide an upper bound for the error of…
The aim of this paper is to show that, for any $p \in [1,\infty)$, the $W^{1,p}$-anisotropic interpolation error estimate holds on quadrilateral isoparametric elements verifying the maximum angle condition ($MAC$) and the property of…
We study the generalization error of functions that interpolate prescribed data points and are selected by minimizing a weighted norm. Under natural and general conditions, we prove that both the interpolants and their generalization errors…
In this paper, we address the problem of interpolation of smooth convex-concave functions. Interpolation is a key step for computer-assisted estimation of worst-case performance via PEP-like techniques, and smooth convex-concave functions…
How good is a triangulation as an approximation of a smooth curved surface or manifold? We provide bounds on the {\em interpolation error}, the error in the position of the surface, and the {\em normal error}, the error in the normal…
Mean value interpolation is a method for fitting a smooth function to piecewise-linear data prescribed on the boundary of a polygon of arbitrary shape, and has applications in computer graphics and curve and surface modelling. The method…
We present the error analysis of Lagrange interpolation on triangles. A new \textit{a priori} error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on…
We discuss the error analysis of linear interpolation on triangular elements. We claim that the circumradius condition is more essential then the well-known maximum angle condition for convergence of the finite element method. Numerical…
We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are…
The main result in this paper is an error estimate for interpolation biharmonic polysplines in an annulus $A\left( r_{1},r_{N}\right) $, with respect to a partition by concentric annular domains $A\left( r_{1} ,r_{2}\right) ,$ ....,…
Interpolation error estimates in terms of geometric quality measures are established for harmonic coordinates on polytopes in two and three dimensions. First we derive interpolation error estimates over convex polygons that depend on the…
We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three dimensional maximum…
We study in this paper the function approximation error of linear interpolation and extrapolation. Several upper bounds are presented along with the conditions under which they are sharp. All results are under the assumptions that the…
We use $L^2$ estimates for the $\bar\partial$ equation to find geometric conditions on discrete interpolating varieties for weighted spaces $A_p(\C)$ of entire functions such that $| f(z)|\le Ae^{Bp(z)}$ for some $A,B>0$. In particular, we…
In this dissertation, it is first shown that, when the radial basis function is a $p$-norm and $1 < p < 2$, interpolation is always possible when the points are all different and there are at least two of them. We then show that…
In CAGD the design of a surface that interpolates an arbitrary quadrilateral mesh is definitely a challenging task. The basic requirement is to satisfy both criteria concerning the regularity of the surface and aesthetic concepts. With…
The Performance Estimation Problem (PEP) approach consists in computing worst-case performance bounds on optimization algorithms by solving an optimization problem: one maximizes an error criterion over all initial conditions allowed and…