Collocation Methods for Parabolic Equations in a Single by Jim Jr.;Dupont, Todd Douglas PDF

By Jim Jr.;Dupont, Todd Douglas

ISBN-10: 3540067477

ISBN-13: 9783540067474

ISBN-10: 3540383379

ISBN-13: 9783540383376

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Extra info for Collocation Methods for Parabolic Equations in a Single Space Variable: Based on C1-Piecewise-Polynomial Spaces

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N L~176 If, in addition, At h + h3 At < c

W - U . 48) = 0([t +]1 ull2, O] IIUllr+2,0 hr + ]lu r-t,2 hr-iA + ]1UHr+t, t hr+l t +[llUllo,s+ 2 + Itullz, s+iJ(At) s+l ), 53 i -53 by the estimates leading to (6. 18). ~, k='l_ 2 ~. 51) holds. ~=t = O(h r + h r - t A t of p. ~176 [o, T]) IIL~~ (6. 50) = t~ ' h ll~JILoo(xx Lo",T]) = O ( h r § hr-Iht + (At)s+i) It then follows f r o m (6. 46), (6. 52) . (h r+l + hr-IAt + (At) s+l) G t ( h , At) . Hence, the following theorem has been proved. 1-54 54 Theorem 6. 16) corresponding to (6~ Assume that a,b, and third derivatives in a neighborhood of the solution c have bounded u and that the n o r m s of u appearing in the definition ( 6.

8 -~(I - J )u can be broken into three terms by use of (6 . 4) , used for 8 {-~- I@(I T r , 6)Su-~-(~ij, t) (I- ~IJ-E II r+ ~, 0hr+i) ' q = 0, and homogeneity in the interval length. - T* s,E Next 8 )u } (x, Tk~ ) : { -~- (I _ T*s, ()u } (x, Tkf) -- o(llull o, s+z(At) by (6. 6), (6~ 9), q = l, and homogeneity. s+l) Also, it can be shown that { ~ t (I-Tr, 6)(~(I-T:, )u}(~ij, Tkf ) = O(IIUl]r_i,Z hr'IAt) as follows. i ' 8r-lu - i , o~) -8xr- ( x i - l + ~ o~ , t) do' . 36 Similarly, for s > l, -- {I| )u )(~,t) I-T* is exact on S~E = 1-36 ~0(1,~).

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Collocation Methods for Parabolic Equations in a Single Space Variable: Based on C1-Piecewise-Polynomial Spaces by Jim Jr.;Dupont, Todd Douglas


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