Mohammed-Najib Benbourhim

A vector spline approximation

We introduce a new family Pα,β of spline minimization problems for vector fields, defined by where V = (u, v) is a two component vector function, X is the Beppo-Levi space D−2L2(R2) x D−2L2(R2), Xi = (xi, yi are the interpolation points, and Vi = (ui, vi) are data values. A coupling between V components is achieved by the divergence (div) and rotational (rot) operators. α, β are fixed real positive constants controlling the relative weight on the gradient of the divergence and rotational fields. The explicit control on divergence and rotational operators is well suited for geophysical fluid flow interpolations; it allows us to cope with the great differences frequently observed in the magnitudes of the divergent and rotational parts of the flow. Through the general spline formalism, existence and uniqueness of the solution is proved. The analytical solution is explicitly calculated and numerical examples are presented. For α (and β) → 0, “limit” problems are defined and their analytical solutions are given.

Approximation of vectors fields by thin plate splines with tension

We study a vectorial approximation problem based on thin plate splines with tension involving two positive parameters: one for the control of the oscillations and the other for the control of the divergence and rotational components of the field. The existence and uniqueness of the solution are proved and the solution is explicitly given. As special cases, we study the limit problems as the parameter controlling the divergence and the rotation converges to zero or infinity. The divergence-free and the rotation-free approximation problems are also considered. The convergence in Sobolev space is studied.

A vector spline approximation with application to meteorology

Abstract We present a new vector approximation based on general spline function theory. The smoothing functional involves the divergence ( divV = ∂ x u + ∂ y v ) and the rotational ( rotV = ∂ x v – ∂ y u ) of the approximated vector field V = ( u, v ). The method can be applied for the restitution of velocity fields from observed data, in fluid mechanics and especially in geophysical fluid flows (horizontal wind fields in meteorology, oceanic currents, etc.).

Pseudo-polyharmonic vectorial approximation for div-curl and elastic semi-norms

Vector field reconstruction is a problem arising in many scientific applications. In this paper, we study a div-curl approximation of vector fields by pseudo-polyharmonic splines. This leads to the variational smoothing and interpolating spline problems with minimization of an energy involving the curl and the divergence of the vector field. The relationship between the div-curl energy and elastic energy is established. Some examples are given to illustrate the effectiveness of our approach for a vector field reconstruction.

Spline elastic manifolds

This paper is intended to give the definition and main properties of a spline elastic manifold which minimizes the stored energy function of an isotropic hyperelastic material with local constraints.

Error estimates for interpolating div--curl splines under tension on a bounded domain

This paper discusses error estimates and convergence for interpolation by div-curl spline under tension of a vector field in the classical vectorial Sobolev space on an open bounded set with a Lipschitz-continuous boundary. A property of convergence is also given when the set of interpolating points becomes more and more dense.

DIV-CURL WEIGHTED MINIMIZING SPLINES

The paper deals with a div-curl approximation problem by weighted minimizing splines. The weighted minimizing splines are an extension of the well-known thin plate splines and are radial basis functions which allow the approximation or the interpolation of a scalar function from given scattered data. In this paper, we show that the theory of the weighted minimizing splines may also be used for the approximation or for the interpolation of a vector field controlled by the divergence and the curl of the vector field. Numerical examples are given to show the efficiency of this method.

Meshless Pseudo-Polyharmonic Divergence-Free and Curl-Free Vector Fields Approximation

In this paper, we propose a meshless approximation of a vector field in multidimensional space minimizing quadratic forms related to the divergence or to the curl of a vector field. Our approach guarantees the conservation of the divergence-free or the curl-free properties, which are of great importance in applications. For instance, divergence-free vector fields correspond to incompressible fluid flows, and curl-free vector fields correspond to magnetic fields in the equations of classical electrodynamics. Our construction is based on the meshless approximation by the pseudo-polyharmonic functions which are a class of radial basis functions. We provide an Helmholtz–Hodge decomposition of the used native functional space. Numerical examples are included to illustrate our approach.

Spline approximant in Hilbertian subspaces of &Ω

SummaryIn this paper we give a new approach of approximation by spline functions. We define and study approximant spline functions which can be easly calculated without solving a linear system. We investigate also the error in using approximant spline functions.