By K. Rektorys
ISBN-10: 9027713421
ISBN-13: 9789027713421
Read or Download The Method of Discretization in Time and Partial Differential Equations PDF
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Extra resources for The Method of Discretization in Time and Partial Differential Equations
Sample text
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. The Method of Discretization in Time and Partial Differential Equations by K. Rektorys
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