UDC 519.63, 519.651 High-Order Vector Nodal Finite Elements with Harmonic, Irrotational and Solenoidal Basis Functions O. I. Yuldashev, M. B. Yuldasheva Laboratory of Information Technologies Joint Institute for Nuclear Research Joliot-Curie 6, 141980 Dubna, Moscow Region, Russia In the present paper a concept of vector nodal finite element has been introduced, algorithms of construction of the vector nodal basis functions with high approximate properties from special functional spaces are presented. <...> Examples of high-order interpolation of harmonic, irrotational vector fields by the developed finite elements illustrate their approximate advantage in comparison with the standard Lagrange elements. <...> Key words and phrases: vector nodal finite elements, harmonic, irrotational, solenoidal basis functions, interpolated polynomials, approximations of high order. 1. <...> Introduction ω ∈ Rn (n=2,3) is a closed subset with a Lipschitzian boundary and with a nonempty set of inner points often called as a cell or a finite element; P is an m-dimensional by different ways such as the finite element method, the volume and boundary integral equation methods etc. <...> One of possible ways to get over the complications is use of special high order approximations. <...> In the present paper we suggest a new class of finite elements for vector-functions approximations with high accuracy. <...> According to the classical definition [4,5], a finite element is a triple (ω,P,Φ), where The finite elements as independent objects may be used for solving elliptic problems space of functions defined on ω (usually this is a space of polynomials); Φ is a set of linearly independent linear functionals Fi : P →R1, i = 1, . . . ,m. <...> In the nodal finite elements Fi(ϕ) is the value of a function ϕ at the node xi ∈ ω. <...> If for a set of functions {Nj}j=1,.,m ∈ P for each j the system of linear algebraic equations Fi(Nj) = δij, i = 1, . . . ,m, is solvable, then any function ϕ ∈ P can be represented in the form ϕ(x) = ∑ Fi(ϕ)Ni(x). i=1 m System (1) is used for finding coefficients in the representation and Ni (1 i m) is called as basis or shape function. <...> Accuracy of interpolation by means of basis functions may <...>