A $C^1 $ Finite Element Collocation Method for Elliptic Equations

Collocation at Gaussian quadrature points as a means of determining a $C^1 $ finite element approximation to the solution of a linear elliptic boundary value problem on a square is studied. Optimal order $L^2 $ and $H^1 $ error estimates are established for approximation in a function space consisting of tensorproducts of $C^1 $ piecewise polynomials of degree not greater that r, where $r \geqq 3$.