1549301742-The_Theory_of_Difference_Schemes__Samarskii

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258 Difference Schemes for Elliptic Equations

4.2 THE MAXIMUM PRINCIPLE


  1. The canonical form of a grid equation of common structure. The maxi-
    mum principle is suitable for the solution of difference elliptic and parabolic
    equations in the space C and is certainly true for grid equations of common
    structure which will be investigated in this section.
    Let w be a finite set of nodes (a grid) in some bounded domain of
    the n-dimensional Euclidean space and let P E w be a point of the grid w.
    Consider the equation


(1) A(P) y(P) = B(P, Q) y(Q) + F(P), PE w,
QEPatt'(P)

related to a function y(P) defined on the grid w. Here the coefficients of
the equation A(P) and B(P, Q) and the right-hand side of the equation
F(P) are given grid functions; Patt'(P) Cw, being the set of all the nodes
of the grid w except for the node P, is the neighborhood of the node P.
The pattern of the grid equation (1) at the node P consists, evidently, of
the node P itself and its neighborhood Patt' ( P).
Similar equations do arise in grid approximations of integral equations.
In what follows we will suppose that coefficients A(P) and B(P, Q) are
subject to the conditions


for all PE w, Q E Patt'(P),
(2)
{

A(P) > 0, B(P, Q) > 0


D(P)=A(P)- L B(P,Q)>O.
QEPatt'(P)

A point P is called a boundary node of the grid w if at this point
the value of the function y( P) is known in advance:


(3) y(P) = μ(P) for P E /,


where I is the set of all boundary nodes.
Comparing (3) and (1) we see that on the boundary I we must set
formally A(P) _ 1, B(P, Q) = 0 and F(P) = μ(P).
We call the nodes, at which equation (1) is valid under conditions (2),
inner nodes of the grid; w is the set of all inner nodes and w = w + I
is the set of all grid nodes. The first boundary-value problem completely
posed by conditions (1)-(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.

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