Farin [
1] used the notion of Bézier
simplices in Sibson coordinates [
2] to
propose a
![$ C^1(\Omega)$](img2.png)
interpolant. The
![$ C^1(\Omega)$](img2.png)
interpolant has quadratic precision,
and reduces to a cubic polynomial between adjacent nodes on
the boundary
![$ \partial \Omega$](img3.png)
. In this
report, we present the
![$ C^1(\Omega)$](img2.png)
formulation and
propose a computational methodology for its
implementation in the context of a Galerkin procedure
for the solution of partial differential
equations (PDEs). The
approach involves the
transformation of the original Bernstein basis functions
![$ B_{ {\mathbf{i}}}^{3}(\Phi)$](img4.png)
to new shape functions
![$ \Psi$](img5.png)
,
such that the shape functions
![$ \psi_{3I-2}$](img6.png)
,
![$ \psi_{3I-1}$](img7.png)
,
![$ \psi_{3I}$](img8.png)
for node
![$ I$](img9.png)
are directly
associated with the three nodal degrees of freedom
![$ w_I$](img10.png)
,
![$ \theta_{Ix}$](img11.png)
,
![$ \theta_{Iy}$](img12.png)
, respectively. Such a
transformation renders the interpolant
amenable to application in a Galerkin scheme for the
solution of fourth-order elliptic PDEs. Numerical
interpolation results have verified the
smoothness and quadratic precision
property of the interpolant, and currently its application to
the biharmonic equation is in progress.