1549301742-The_Theory_of_Difference_Schemes__Samarskii

694 Methods for Solving Grid Equations

7. Difference schemes for elliptic equations of general form. As we have
   mentioned above, the applications of ATM in the numerical solution of an
   operator equation of the first kind consist of several steps:

• determination of the operator R;


• design of special "triangle" operators R 1 and R 2 ;


• calculations of the equivalence constants c 1 and c 2 and b, D.;


• selection of the number n = n( c) of the iterations and a set of
  parameters { r;},
  We begin our exposition with a discussion of examples that make it
  possible to draw fairly accurate outlines of the possible theory regarding
  these questions and with a listing of the basic results together with the
  development desired for them. C0111mon practice involves the Laplace op-
  erator as the operator R in the case of difference elliptic operators A. The
  present section is devoted to rather complicated difference problems of the
  elliptic type, Here and below it is supposed that the domain of interest is
  a p-dimensional parallelepiped G = { 0 < x °' < l°', CY = l, 2, ... , p} with
  the boundary r (a rectangle for the case p = 2), on which the boundary
  condition of the first kind is imposed:

vlr = μ(x),

In what follows it is required to find a continuous m G solution of the

Dirichlet problem

(50) Lv=-f(x), xEG, ulr=p(x),

with a second-order elliptic operator L involved.

vVith this aim, we introduce in the domain (; the usual grid

with the boundary ''ih, so that ::.ih = wh U fh. As usual, an inner product in

0

the space H = D of all grid functions defined on wh and vanishing on the

boundary ih is taken to be

(y, v) = 2= y( x) v( x) h 1 h 2. • hp.

Wh

It seems worthwhile giving several particular cases.