1549301742-The_Theory_of_Difference_Schemes__Samarskii

276 Difference Schemes for Elliptic Equations In a similar way we find that etc. By means of the method of separation of variabl ...

Some properties of difference elliptic operators 277 (12) and (13) with (14) [y, v] = (y, v) + hi h2 [ 4 y(O, 0) v(O, 0) + y(O, ...

278 Difference Schemes for Elliptic Equations We list below the properties of the difference Laplace operator acting from 0 0 rl ...

Some properties of difference elliptic operators^279 The operator A admits the bilateral estimate (18) where f; is the sarne a ...

280 Difference Schemes for Elliptic Equations where ACYy= -A-;,y= -2yxu/hCY ACY Y = -Aty = 2 Yxa/hCY for for x CY = (^0) l With ...

Some properties of difference elliptic operators 281 The sides of rectangles, which constitute the domain Gare assumed to be com ...

282 Difference Schernes for Elliptic Equations is valid with 12 M - o o-2 lfT' V 'I '2 Proof Suppose that vk, k 2 ( x) and Ak, k ...

Some properties of difference elliptic operators 283 With this in n1ind, it remains only to evaluate the ren1aining sums Ni-I N2 ...

284 Difference Schemes for· Elliptic Equations where ko:(x) are sufficiently smooth functions, 0 < c 1 < ko:(x) < c 2 , ...

Some properties of difference elliptic operators 285 for any v E H, where 0 0 2 A Y =-A Y = - L Yx" Xa a:=! and the constants f; ...

286 Difference Schen1es for Elliptic Equations where E stands for the identity operator. The self-adjointness of the operator A ...

Some properties of difference elliptic operators 287 In this line, the operator Laf3 u is approximated by the difference op- era ...

288 Difference Schemes for Elliptic Equations for v = k 12 u,, 2 yields Likewise, from which estimates (33) follow in1mecliately ...

Some properties of difference elliptic operators 289 Let us investigate the properties of the operator Ay = -Ay for any 0 y E rl ...

290 Difference Schemes for Elliptic Equations The expression for (A-y, y) can be transformed without difficulties 0 in just the ...

Higher-accuracy schemes for Poisson's equation 291 7 ,,._ ____^4 ......., 6 3 <)----------""('+---------~ 1 s^4 5 Figure 16. ...

292 Difference Schemes for Elliptic Equations assuming that v( x) has a continuous second derivative on the segment [x-h,x+h]; ( ...

Higher-accuracy sche1nes for Poisson's equation 293 Because of this, the equation A'y = -<p, (7) provides an approximation of ...

294 Difference Sche1nes for Elliptic Equations Estimation of a solution of the difference boundary-value problem. Con- sider no ...

Higher-accuracy schemes for Poisson's equation holds. This implies that Scheme (9) is of fourth-order accuracy when u E C(^6 ), ...