Inner nodes

Fig. 8. Diffraction space according to the "disordered stacking model" (a) achiral (zigzag) tube (b) chiral tube. The parallel circles represent the inner rims of diffuse coronae, generated by streaked reflexions. The oo.l nodes generate sharp circles. In (a) two symmetry related 10.0 type nodes generate one circle. In the chiral case (b) each node generates a separate corona [9].

F are known as boundary nodes. The set of all boundary nodes is denoted by 7/j. In Fig. 3 the boundary and inner nodes are quoted with the marks and o, respectively. 

As can readily be observed, there are boundary nodes with the distance to the nearest inner nodes smaller than /Zj or /z,. In spite of the obvious fact that the grid in the plane is equidistant in Xy and Xj both, the grid = W/j - - 7 , in the domain G is non-equidistant near the boundary. This case will be the subject of special investigations in Chapter 4. 

The difference operator L,. defined at all inner nodes of the grid... 

Let h be a vector parameter related to the distribution density of the nodes of the grid and let and 7/, be the sets of its inner and boundary nodes. With these ingredients, the difference problem... 

Let us obtain this a priori estimate by multiplying equation (51) by and summing over all grid nodes ofw ,. In terms of the inner products the resulting expression can be written as... 

Let be the set of grid functions defined at the inner nodes of the grid W/j. The set so constructed is certainly linear. Once equipped with the inner product (y, v) = Vi h and associated norm y = / y, y),... 

We have written the difference equation (14) at a fixed node x = x. With an arbitrarily chosen node it is plain to derive equation (14) at all inner nodes of the grid. Since at all the nodes x, i = 1, 2,.. ., IV — 1, the coefficients a, and are specified by the same formulae (15), scheme (14)-(15) is treated as a homogeneous conservative scheme. Because of this, we may omit the subscript i in formulae (14)-(15) and write down an alternative form of scheme (14) ... 

Following established practice, at each of the inner nodes 6 we compose a five-point regular cross pattern, whose nodes a = 1,2,... 

As a final result problem ( ) is associated with the Dirichlet difference problem relating to the determination of a grid function y x) defined on the grid W , satisfying at the inner nodes, that is, on the equation... 

We offer below more a detailed classification of inner nodes. With this aim, let us draw up a straight line parallel to the axis Ox through an inner node X Its intersection with the domain G is an interval (or several... 

The grid in view is supposed to be connected, it being understood that any two inner nodes can be joined by a polygonal line, the parts of which are parallel to the coordinate axes and vertices coincide with inner nodes of the grid. Then at least one of the four nodes a = 1,2, of... 

At each of the inner nodes k 6 we approximate the difference operator... 

We call the nodes, at which equation (1) is valid under conditions (2), inner nodes of the grid uj is the set of all inner nodes and ui = ui + y is the set of all grid nodes. The first boundary-value problem completely posed by conditions (l)-(3) plays a special role in the theory of equations (1). For instance, in the case of boundary conditions of the second or third kinds there are no boundary nodes for elliptic equations, that is, w = w. 

In the case of the difference scheme for the Dirichlet problem (24)-(26) of Section 1 the definition (4) of connectedness coincides with another definition from Section 1. The very definition implies that the point P may be boundary and, hence, the connectedness is to be understood that every point of the boundary belongs to the neighborhood Patt [P) of at least one inner node. 

Theorem 1 (the maximum principle) Let y P) const be a grid function defined on a connected grid w and let both conditions (2) and (4) hold. Then the condition Cy P) < 0 (C y P) > 0) on the grid w implies that y(P) cannot attain the maximal positive (minimal negative) value at the inner nodes P E u>. 

Other ideas are connected with the operator A, which coincides with A at the near-boundary nodes and with A at the remaining inner nodes, leading to an alternative form of writing... 

Let Qh be the space of all grid functions defined at the inner nodes... 

The starting point in the further development of the difference scheme is the approximation of the elliptic operator Aw. We learn from Section 1 of Chapter 4 that at all of the inner nodes Aw Aw for x G... 

These equations are written at all inner nodes + = of the grid uif and for all / = / > 0. Let us stress here that the first scheme is implicit along the direction and it is explicit along the direction while the second one is explicit in the direction and it is implicit in the direction ajg. Equations (9)-(10) are supplemented with the initial conditions... 

---