Exercice 3. We consider the following boundary value problem: 2 P{(x)+= f(x), €10,1[, u(0) u(1)=0 For solving the problem P by a finite difference method we use the tree points scheme, for an integer N≥2 24-1-24+4+1 = ih, h= h2 N+1 u" (2)≈ and the centered scheme u'(x)= U-1-Ui+1 2h 1) Provide the finite difference discrete scheme of the problem P and write it as a linear system A, ub by specifying the matrix A, and the vector bh- = =(1,..., IN), prove 2) Let y=A, where y = (1,..., VN) and x = 20 then x 20 for iЄ (1,2,.....N}. Deduce that the matrix A, is invertible. 3)We consider the following function 8(x) = −(1+1)² Ln(1+1) + +2x) Ln(2), E [0,1]. Prove that the function is a solution of the problem P when f=1. - 4) We define the vector 0 = ((x1), 0(x2),..., 6(N)), prove that there exists a constant C independent of h such that |(A, 0h) – 1| <Ch*, i€{1,2,N}. 5) Deduce from the above that there exists a constant M independent of h such that AM, where ||A|=suplay)- 6) Prove that the finite difference scheme is convergent with an order equal to 2.