A VECTOR SPLINE APPROXIMATION

We introduce a new family Pα,β of spline minimization problems for vector fields, defined by {A figure is presented} 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. © 1991.