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Numerical_Methods for Nonlinear Variable Problems Episode 11

Tham khảo tài liệu 'numerical_methods for nonlinear variable problems episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 387 it follows from 4.301 and 4.297 4 that the boundary integral in 4.300 reduces to f AVu -npdr Jri where Ti x x xv x2 0 Xj 1 x2 0 implying in turn that 4.300 reduces to f AVm Vỉ dx f fv dx f AVu nr dr. 4.302 Jii Jf Jti Combining 4.297 5 and 4.302 and using the second relation 4.301 we obtain after integrating by parts over r I f AVu Vv dx f fc x1 - x1 0 Xj 0 dxj Jii Jo dXi ỠXi f fvdx f grv dr. 4.303 Jo Jfi Conversely it can be proved that if 4.303 holds for every V e tC where C r IV e CZ Q v 0 x2 p l x2 if 0 x2 1 V 0 in the neighborhood of r0 4.304 where r0 x x 6 x1 x2 0 xx 1 x2 1 then u is a solution of the boundary-value problem 4.297 . Relations 4.301 4.303 suggest the introduction of the following subspaces of H1 V r p 6 H Q 11 0 x2 1 1 x2 a.e. 0 x2 1 d dXiXxj 0 e L2 0 1 4.305 Kj t IV 6 V v xỵ 1 0 a.e. 0 Xj 1 . 4.306 Suppose that V is endowed with the scalar product v w y v w Hi fi -J v Xi 0 w x1 0 dxj 4.307 Jo dxr dXỵ and the corresponding norm IK V 2. 4.308 We then have the following Proposition 4.21. The spaces V and vo are Hilbert spaces for the scalar product and norm defined by 4.307 and 4.308 respectively. Moreover the seminorm X1 0 ax-i defines a norm equivalent to the V-norm 4.308 over pQ. 388 App. I A Brief Introduction to Linear Variational Problems Exercise 4.11. Prove Proposition 4.21. We now define a bilinear form a V X V - R and a linear functional L V- R by a y w f ÃVtt Vw dx f k x1 v x1 ữ - w x1 ơ dx1 Jq Jo IX ỵ uXỵ 4.309 L p ĩfvdx f g2vdr 4.310 Jn Jr respectively. We suppose that the following hypotheses concerning A k f 9i hold mW 01eL2 ri 4.311 k e L 0 1 k x2 a0 0 a.e. on 0 1 4.312 A satisfies 4.47 . 4.313 From the above hypotheses we find that a . is bilinear continuous over V X V and vo-elliptic and that L . is linear continuous over V we can therefore apply Theorem 2.1 of Sec. 2.3 to prove Proposition 4.22. If the above hypotheses on A k f g2 hold and