hw8 - gives the same result I is the identity operator Iu i...

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Math 128a, fall 2011, Chorin, theory homework 8, due the week of Nov. 6. 1. Consider a vector u = ( u 1 ,u 2 ,...u N - 1 ) living at the points ih,i = 1 ,..., ( N - 1) , it being understood that u 0 = u N = 0. Define the operators D + ,D - ,S by the following rules: ( D + u ) i = ( u i +1 - u i ) /h, ( D - u ) i = ( u i - u i - 1 ) /h, ( Su ) i = u i +1 . Check the following identities:(i) D + = SD - ; (ii) D + D - u = ( S + S - 1 - 2 I ) u/h 2 ; (iii) D + = ( S - I ) /h , where I is the identity. Rep- resent each of D + ,D - ,S,D + D - by a matrix; check that the matrix that represents ( D + D - ) is indeed the product of the matrices that represent D + and D - . Be careful about what happens at i = 1 ,i = N - 1. In (ii), an expression such as ABu where A,B are operators and u is a vector means “ A applied to the vector Bu ”; you peform B first and A second. Two operators are equal if mutiplying an arbitrary vector by either one
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Unformatted text preview: gives the same result. I is the identity operator: ( Iu ) i = u i . 2. Show that u i = ∑ j = i j =1 h ( D-u ) j . Deduce that || u || ≤ || D-u || , where || u || 2 = ∑ i = N-1 i =1 u 2 i h. 3. Show that if the function f is twice continuously differentiable, and if the equation-y 00 = f,y (0) = y (1) = 0 , is approximated by (-D + D-) u = f,u = u N = 0 , , then the norm || y-u || of the error vector with compo-nents y i-u i equals C ( x ) h 2 + O ( h 4 ), where C ( x ) is independent of h . 1...
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