Proportional errors Interpolation

A finite element method based on these functions would have an error proportional to Ax2. The finite element representations for the first derivative and second derivative are the same as in the finite difference method, but this is not true for other functions or derivatives. With quadratic finite elements, take the region from x,.i and x,tl as one element. Then the interpolation would be... 

The effect of assuming a particular activation energy is thus insignificant for the interpolative aspect of Equation (15). For extrapolation to other reactions, however, there will be an error in the estimated critical temperature rise which is proportional to and opposite in direction to the error in the assumed activation energy of hydrotreating. 

We use our model, as defined by the Hamiltonian, to interpolate between the electronic energies of the hydrogen atom Eh and the proton as a function of the solvent coordinate q. The electronic energy of the proton is 0, and it corresponds to an equilibrinm valne of = -1 in our normalization of q. For a given distance, we calculate the electronic energy E(q = 0) of the hydrogen atom from our model for <7 = 0 from the first part of Equation (3.16). The difference AE(q = 0) contains the multibody effects not contained in our model. We assume that this error is proportional to the occupancy ... 

Interpolation in Time. The basic idea is to compute accurate approximations of solution derivatives at the endpoints of each integration interval by extrapolation of divided differences of appropriate solution approximations. Properties and a general implementation of this approach have been studied in [7]. For an application in the adaptive MOL context the basic algorithm must be modified, e.g. as the space grid may change from step to step. Assume that K lines of the time extrapolation tableau have been computed in the step, i.e. the error of the highest extrapolated solution approximation at tji is proportional to If we require an interpolation error of at... 

Several approaches have been proposed in the literature to estimate the overall heat transfer coefficient [7-12]. The most common approach implemented in commercial lab scale calorimeters (like the RCl calorimeter from Mettler-Toledo), uses a two point calibration (nsing a calibration heater of a known power) [13]. The calibration is carried out at the beginning and at the end of the polymerization (in absence of reaction) and then the value of U is interpolated. There are two main drawbacks with this approach The first is that the heat of reaction cannot be obtained online because the expected change of U during the polymerization reaction (typically a decrease) cannot be calculated until the end of the reaction, and the second drawback is that the off-line calculated depends on the interpolation of the U values calculated at the beginning and at the end of the experiment. Although commercial equipment allows for different interpolations methods (linear, proportional to conversion, etc.), significant errors can be made in the computation of the... 

---