1549301742-The_Theory_of_Difference_Schemes__Samarskii

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Difference Schemes for Elliptic Equations This chapter is devoted to various difference approximations of second-order elliptic ...

238 Difference Schen1es for Elliptic Equations solving the Poisson equation p u ,:i2 'U /:,.u = 2= ax2 = -t(x), a=l ex (1) x E G ...

The Dirichlet difference problem for Poisson's equation 239 2 3 0 1 4 Figure 6. The regular "cross"-pattern which is defined on ...

240 Difference Schernes for Elliptic Equations if v(x) is an arbitrary function having no less than four derivatives with respec ...

The Dirichlet difference problem for Poisson's equation 241 2 3 0 1 4 Figure.^8. Th e. uregu^1 ar " cross " pattern where h 1 ± ...

242 Difference Schemes for Elliptic Equations If, for instance, h 1 - = h 1 + = h 1 , then A~ v = A1 v = vx, x,, etc. The proced ...

The Dirichlet difference problen:1 for Poisson's equation 243 domain. We follow the second way of approximating the Laplace oper ...

244 Difference Schemes for· Elliptic Equations By the same token, UN~ II J \* UN= -^1 h With these relations established, we arr ...

The Dirichlet difference problem for Poisson's equation 245 x 2 , ' 2 , '' ' ( il hl 'i2h2) " , " ,, h 2 " , " " ' , " ' " ' " , ...

246 Difference Schernes for Elliptic Equations The nodes on its boundary (for i 1 = 0, N1 or i 2 = 0, N2) except for four points ...

The Dirichlet difference problem for Poisson's equation 247 It is easily seen from ( 19) and ( 20) that no values ofμ( x) at the ...

248 Difference Schemes for Elliptic Equations This scheme provides the local approximation of order 1 since 1/Ji =(Au+ J(x)), = ...

The Dirichlet difference problem for Poisson's equation 249 intersection of the straight lines x~") = iexhex, CY= 1, 2 with the ...

250 Difference Schemes for Elliptic Equations wh I ()' be the set of all near-boundary nodes with respect to Xa and wh l ()' be ...

The Dirichlet difference proble111 for Poisson's equation 251 a) b) Figure 12. a) A disconnected grid. b) A connected grid. Ou ...

252 Difference Schemes for Elliptic Equations where h: is the distance between the nodes x and x(-la), or A * -y -1 ( y( + l") - ...

The Dirichlet difference problem for Poisson's equation 253 2 1 h l (^0) h1 h2 4 a) a) A; y = !: ( Y1 - Yo h1 h1 b) Aj y = !: ( ...

254 Difference Schemes for Elliptic Equations where 1jJ is the approxirnation error equal for cp(x) = f(x) to 1/;=Au+cp=Au-Lu at ...

The Dirichlet difference problem for Poisson's equation 255 Remark Quite often, the Dirichlet problem is approximated by the met ...