Adjacent elements, support points

There are really many different variants of this method based on the selection of the support points and the elements to discretize the interval. Some of them have special names that highlight their approach. For instance, if the points are selected as the roots of an orthogonal polynomial and if the elements have only one point in common, the method is said to be a finite-element orthogonal collocation. On the other hand, if each element consists of three points and the adjacent elements share two points, the method is said a finite-difference method. In some cases, when the elements have common points, the single residual is not zeroed, but the sum of residuals is calculated in the same point using all the elements that are sharing it. The aim of these variants is to find a well-conditioned system of equations with a structure that makes its solution particularly efficient when the number of variables is rather large. 

The adjacent elements have two or more common support points. 

The adjacent elements have one single common support point... 

In the first case, the two adjacent elements are partially overlapped, whereas in the second case the last support point of the element coincides with the first support point of the next element... 

On the other hand, if two adjacent elements have P -1 common support points, the series is Xi = Xa,X2,. .., xp belonging to the first element. 

Figure 6.3 Support points of adjacent elements with two common points.

Third strategy The adjacent elements have one single common support point and in this point, the adjacent elements have the same first derivative. 

The method of finite differences enters the family of the first strategy with the overlapping of two points between adjacent elements and the use of the first alternative (elements with different size) with one single internal support point (three points for each element). 

On the other hand, if we use the first strategy and several points are shared by the adjacent elements, the only acceptable alternative is to have a limited number of support points for each element (which could be of different size), but efficiently selected. 

Let us now consider the third strategy. The adjacent elements have one single common support point and in this point the adjacent elements have the same first derivative. In this case also, it is possible to find the internal points such that the resulting polynomial (which exploits the two derivatives at the extremes in this case) minimizes the maximum polynomial error function. For instance, in the case of sixth-order polynomial valid in the interval... 

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