5. (Yet another proof of second order accuracy of central difference scheme for Poisson equa- tion in 1D) Consider the 1
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5. (Yet another proof of second order accuracy of central difference scheme for Poisson equa- tion in 1D) Consider the 1D Poisson equation U" = f on [0, 1], which can be solved numerically by Uj+1 − QUj + Uj−1 h² = fj, uo = Un+1 = 0. Now, we would like to provide another simple error estimation for this 1D problem. (uj — uj-1)/h. It is easy to see that (2) can Let D+Uj be written as D+D_uj (uj+1 − uj)/h and D_u¡ f. The exact solution (assuming to be in C4([0, 1])) satisfies D+D_U(xj) = U"(xj)+9jh² = fj+9;h² for some g; which can be assumed to be uniformly bounded with respect to j. Let ej = U(xj)-uj. Then the error equation is D+D_ej = gjh². = U(0) = U(1) = 0, = (1) (2)
(d) Consider any vector (vo, V₁, Un+1). Assuming vo= 0, prove the following inequality which is a discrete version of the Poincare inequality (https://en.wikipedia.org/ wiki/Poincare_inequality): for any k € {0, 1, 2,...,n+1} |vk| ≤ ||D+V ||h₂ where ||D+v|| defo|D+v₁|²h. [Hint: you can start from |vk| ≤ Σ; |V;+1 − vj] and then use Cauchy-Schwarz inequality.] [4 marks] (e) Consider the error equation D+D_e; = gjh². Multiply it by -e;, time it by h, and then sum on j. Prove that ||D+e||h≤Ch² 5 for some universal constant C. Then prove that mạx|ej|<Ch?. [4 marks]