1549301742-The_Theory_of_Difference_Schemes__Samarskii

248 Difference Schemes for Elliptic Equations

This scheme provides the local approximation of order 1 since

1/Ji =(Au+ J(x)), = (ux 1 x 1 + u,; 2 i: 2 + J(x))i

2 83

= ~ L ( h~a+l) - h~a)) ox~ + O(I n 12)

a=l a

= o (I 11 I) , I n 12 = n; + n;.

However, by analogy with Chapter 3, Section 4, 1/Ji is representable by

2

1/Ji = L (7la)i:,i + O(I nl

2

),

a=l

h2 a

7Ja = 6

From such reasoning it seems clear that scheme (22) provides an approxi-

mation of second total order in the negative norm.

4. The Dirichlet difference problem in a domain of rather complicated
   configuration. If a solution of the Dirichlet proble1n needs to be determined
   in a domain G with a nonlinear boundary, the grid wh(G) is, generally
   speaking, non-equidistant near the boundary. We describe below such a
   grid and give the possible classification of its nodes.
   Consider an arbitrary finite domain G with the boundary r in a p-
   dimensional space, where a point with coordinates x 1 , x 2 , ... , xP is denoted
   by x = (x 1 , x 2 , ... , xp)· We proceed to construct a grid in the domain
   G = G + r and confine ourselves, for the sake of si1nplicity, to the case of
   a two-dimensional domain (p = 2). Here a constructive supposition about
   the shape of the domain G is taken into account that the intersection of
   the domain G with any straight line passing through an inner point x E G
   in parallel to the coordinate axis Ox a (ex = 1, 2) consists only of a finite
   number of intervals.
   If the origin of coordinates is inside the domain G, we draw up two
   families of uniform· straight lines

X(ii) l =; "l h l ' ; "l --^0 ' ±1 ) ±2 ' ... ) X 2 (i^2 ) -- ; "2 ,, "2 > Z2 ' =^0 > ±1 , ±2 • • • ,

where h 1 > 0 and h 2 > 0 are fixed numbers. Such straight lines split up
the plane ( x 1 , x 2 ) into rectangles of sides h 1 and h 2. The vertices of these
rectangles with coordinates x 1 = i 1 h 1 and x 2 = i 2 h 2 are called nodes and
the set of all nodes is known as a grid in the plane (x 1 , x 2 ). The nodes
X; = ( i 1 h 1 , i 2 h 2 ) lying inside the domain G refer to inner nodes with the
notation w h = { x; E C} for the set of all such nodes. The points of the