then the element u defined by (4) is a

biharmonic function which solves the following problem:

Then, the function [u.sup.k.sub.1](x, y) + i[v.sup.k.sub.1](x, y) is a

biharmonic function that meets equation coinciding with (32) and the function [w.sup.k.sub.1](x, y) is simply harmonic.

Biharmonic function w in this case has the sense of Love stress function, through which displacement components [u.sub.r], [u.sub.z] and stress components [[sigma].sub.rr], [[sigma].sub.rz], [[sigma].sub.[theta][theta]], [[sigma].sub.zz] can be expressed as [10]

A discrete

biharmonic function v belongs to [V.sub.l]if and only if

Let F be either a harmonic function in D or a

biharmonic function in D which is not harmonic.

Note that a discrete

biharmonic function w = Hw is uniquely defined by the values of all degrees of freedom at nodes in [bar.[GAMMA]].

In particular, [v.sup.[GAMMA].sub.j] is a discrete harmonic function on [[OMEGA].sub.j] in the second order case and a discrete

biharmonic function in the fourth order case.

Using formula (14) and the equality [[phi].sub.1] = [DELTA][[PHI].sub.2], the harmonic and

biharmonic functions and also the metaharmonic function contained in (27) are represented in the circular disc D as series [19, 20]:

Vanderbei, "Probabilistic solution of the Dirichlet problem for

biharmonic functions in discrete space," Annals of Probability, vol.

These radial

biharmonic functions suggest the construction of the following thinplate splines.

Such formulae were recently obtained in [8] for

biharmonic functions which describes elastic materials for a circular multiply-connected domain.

Then let [V.sub.[gamma] s, k] [subset] [V.sup.h] be a space of piecewise discrete

biharmonic functions that may have nonzero degrees of freedom only these which are associated with [[gamma].sub.s, i] i.e., values at vertices and values of normal derivative at midpoints which are on this open master.