Local quasi-interpolation by cubic C1 splines on type-6 tetrahedral partitions

^** Email: sorokina{at}math.uga.edu^*** Corresponding author. Email: zeilfeld{at}euklid.math.uni-mannheim.de We describe an approximating scheme based on cubic C¹ splineson type-6 tetrahedral partitions using data on volumetric grids.The quasi-interpolating piecewise polynomials are directly determinedby setting their Bernstein–Bézier coefficientsto appropriate combinations of the data values. Hence, eachpolynomial piece of the approximating spline is immediatelyavailable from local portions of the data, without using prescribedderivatives at any point of the domain. The locality of themethod and the uniform boundedness of the associated operatorprovide an error bound, which shows that the approach can beused to approximate and reconstruct trivariate functions. Simultaneously,we show that the derivatives of the quasi-interpolating splinesyield nearly optimal approximation order. Numerical tests withup to 17 x 10⁶ data sites show that the method can be used forefficient approximation.