By Aleksandr A., and Evgenii S. Nikolaev Samarskii
From the interior flap: "Since its booklet within the Soviet Union ten years in the past "Numerical equipment for Grid Equations" has develop into a vintage. Revised in 1987 to incorporate new advancements, it really is crucial studying for each pupil and expert who goals at sophisticated and whole wisdom of this subject."
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Extra info for Numerical Methods for Grid Equations: Volume II Iterative Methods
F. 1z . smh z sinhw(z) Vo = -----'--<--::--:- cosh z cosh w( z) Since for fixed f. dw = d(sinhz - z coshz) < 0 dz sinh2 z -, we have dvo _ cosh z ~w - dz sinh z sinh w cosh w 2 2 < cosh z cosh w - % o. Consequently, Vo is maximal for z = O. This gives the estimate Vo = tanh( d). The inequality (35) is proved. Suppose now that Yi is an arbitrary grid function. Vi = 0, f. VN = 2 -h,' The solution of problem (40) is the function o~ i where a is defined above. ~ N, (40) 38 Chapter 5: The Mathematical Theory of Iterative Methods From this we find that (41) We shall bound this expression from above for any h.
Suppose that the operator C map8 the closed set T into it8elf and commute8 with the operator B satisfying the contractive mapping condition. Then a fixed point of the operator B i8 a fixed point (p08sibly not unique) of the operator C. In particular, if 80me power Bn of the operator B is a contractive mapping, then a fixed point of the operator Bn is al80 a fixed point (unique) of the operator B. 20 Chapter 5: The Mathematical Theory of Iterative Methods We turn now to the solution of equation (9) with a non-linear operator A.
L = -cp(x), x E w, Ot,P=1 (74) y(x) = g(x), We write (74) in the form of the operator equation (59), defining the operator A in the usual way: Ay = -Ay, where Y E H(w), YE iI and y(x) = y(x) for x E w. Here the right-hand side f only differs from the right-hand side cp in equation (74) at the near-boundary nodes. To find an explicit formula for f we write out the difference equation at a boundary node, use the boundary condition and move the known values of y( x) on , to the right-hand side of the equation.