Quasi-interpolation by quadratic piecewise polynomials in three variables

A quasi-interpolation method for quadratic piecewise polynomials in three variables is described which can be used for the efficient visualization of gridded volume data. We analyze the smoothness properties of the trivariate splines. We prove that the splines yield nearly optimal approximation order while simultaneously its piecewise derivatives provide optimal approximation of the derivatives of smooth functions. The constants of the corresponding error bounds are given explicitly. Numerical tests confirm the results and the efficiency of the method.