Download PDF by K. Rektorys: The Method of Discretization in Time and Partial

By K. Rektorys

ISBN-10: 9027713421

ISBN-13: 9789027713421

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Extra resources for The Method of Discretization in Time and Partial Differential Equations

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143) is a characteristic property for the functions from W2'\G). 143) is characteristic at least for A: = 1 and k = 2. (Thus, for example, for fc = 1 we have {v; ve W^^\G), f = 0 on F in the sense of traces} = Wi^\G). 5. ) Let G be a bounded domain in Ef^ with a Lipschitz boundary. Then there exist such positive constants kj, fcj (depending only on G, thus not depending on the functions u(x) 17 ) Thus functions with compact support in G. ,||2 < l- V il"l|H'2

As seen from the following examples, it need not be the case in general, and the functions v e Kare to be considered instead of (p e CQ^HG) in order to take the given boundary conditions into account. ^^) On the construction of such a function (g{S) being given) see [3], Chap. 11. On existence questions see [3], Chap. 46. ^®) It is shown ([3], Chap. 33) that the weak solution dees not depend on the choice of the function w{x), the trace of which the function g{S) is. 6. 169) — = h on r dv with/eL2(G), heL^Xr).

29, can be utilized. 113) which belong to D^. Let i/„ = ajt'i + ... 102)). 114) 34 Chapter 2 only, Au„ is not equal to / , in general. Let Au„=^g. 115) A simple calculation ([3], Chap. 116) or, using ||M„ — UQ\\^ ^ C\\u„ — UQ\\, the inequality \\u„ - «ol| ^ "'^""^r^'l . 117) These are the desired error estimates. e. to get such an estimate for C that C be as large as possible). For this reason, great attention is paid in [3], Chap. 18, to appropriate estimates of constants in the so-called inequalities of the Friedrichs type on the base of which it is possible to obtain appropriate estimates for the constant C.

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The Method of Discretization in Time and Partial Differential Equations by K. Rektorys

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