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.
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 
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  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.