Interpolation operators on a triangle with one curved side

Abstract

We construct certain Lagrange, Hermite and Birkhoff-type operators, which interpolate a given function and some of its derivatives on the border of a triangle with one curved side, as well as some of their product and Boolean sum operators. We study the interpolation properties and the order of accuracy (degree of exactness and precision set) of the constructed operators, respectively the remainders of the corresponding interpolation formulas. Finally, we give some numerical examples.

Extension of Some Positive and Linear Operators to Domains with Curved Sides

“…In order to match all the boundary information on a curved domain (as in Dirichlet, Neumann or Robin boundary conditions for differential equation problems), there were considered interpolation operators on domains with curved sides (see, e.g., [4], [6], [9]- [12], [15], [16], [20], [21], [23]). Approximation operators on polygonal domains with some curved sides have also important applications especially in finite element method for differential equations with given boundary conditions and in the piecewise generation of surfaces in computer aided geometric design.…”

Interpolation Operators on Some Triangles with Curved Sides

“…3) Regarding the remainders R L i F, i = 1, 2, 3, of the interpolation formulas F = L i F + R L i F, i = 1, 2, 3, we have: Theorem 1 ( [14]). If F ∈ B 11 (0, 0) then…”

“…2) The orders of accuracy: (19) dex(P H 12 ) = 2, pres(P H 12 ) = {1, x, y, x 2 , xy, y 2 , x 2 y, xy 2 }. For the remainder of the corresponding interpolation formula, F = P H 12 F + R HP 12 F, we have: 14]). If F ∈ B 12 (0, 0) then the following inequality holds…”

“…For the remainder of the interpolation formula, F = S H 12 F + R HS 12 F, we have: 14]). If F ∈ B 12 (0, 0) then the following inequality holds…”

“…The aim of this survey is to present some interpolation and Bernsteintype operators for functions defined on triangles with one or all curved sides (see [11], [12], [14], [15]). They come as an extension of the corresponding operators for functions defined on triangles with all straight sides (see, e.g., [3]- [6], [8]- [10], [13], [23], [24], [26]- [29], [32]).…”