1549301742-The_Theory_of_Difference_Schemes__Samarskii

256 Difference Schemes for Elliptic Equations In the case corresponding to Fig. 13.c we deduce that where nl = ~(h~- + h~+) and ...

The Dirichlet difference problem for Poisson's equation 257 quantities A(x) and B(x,~) are the available coefficients of the equ ...

258 Difference Schemes for Elliptic Equations 4.2 THE MAXIMUM PRINCIPLE The canonical form of a grid equation of common structu ...

The maximum principle 259 _The grid w is taken to be connected, that is, for fixed points P E w and P E w there always exists - ...

260 Difference Schemes for Elliptic Equations On the gridwhr = {(x; = ih,tj = jr), i = 0, 1, ... , N, hN = 1, j = 0, 1, ... } th ...

The maximum principle 261 Since y(P) > y(Q) for all Q E Patt'(P), we find by virtue of the relations D(P) > 0 and y(P) > ...

262 Difference Schemes for Elliptic Equations Corollary 2 The homogeneous equation (1) subjecl to the boundary con- dition (9) £ ...

The maximum principle 263 Corollary For a solution of the problem (13) Ly= 0 on w, y(P) = μ(P) on /, the estimate (14) max I y(P ...

264 Difference Schemes for Elliptic Equations Proof Let a majorant Y(P) be taken such that £ Y = I F(P) I, YI,, = 0, Y(P) > 0 ...

The maximum principle 265 Theore1n 4 Let the conditions D(P) = 0 0 for P E w, D(P)>O for PEw* hold, where~ is a connected gri ...

266 Difference Schemes for Elliptic Equations where Aa Y = Y"' •"a x· a ' p A*y= L A~y, a=l Other ideas are connected with the o ...

Stability and convergence of the Dirichlet difference proble1n 267 has been derived in Section 2 and may be useful in such a set ...

268 Difference Schemes for Elliptic Equations p where e = p-^1 L BC,. Here eQ = 1 if a node x E w;, is regular along the a=l dir ...

Stability and convergence of the Dirichlet difference problem 269 In this case the function D( x) takes on the value D(x) = A(x) ...

270 Difference Schemes for Elliptic Equations we establish the relations 1 1 p 2 A( x) = h a J la+ + -h2 a + 2= h2 (3 ' (3= l (3 ...

Stability and convergence of the Dirichlet difference problem 271 The uniform convergence and the order of accuracy of a differ ...

272 Difference Schemes for Elliptic Equations 4.4 SOME PROPERTIES OF DIFFERENCE ELLIPTIC OPERATORS In this section we reveal som ...

Some properties of difference elliptic operators 273 Observe that ih and, hence, wh contain no nodes (0, 0), (0, N 2 ), (N 1 , 0 ...

274 Difference Schemes for Elliptic Equations where {μk, (x 1 )} is an orthonormal system of eigenfunctions: (μk, , μk;) i = bk, ...

Some properties of difference elliptic operators 275 in the sense of the inner product N 1 -1N 2 -1 (7) (y,v)= LL y(i 1 h 1 ,i 2 ...