Consider the boundary value problem with Dirichlet conditions d²u = sin(x) dx² u(0) = 1 u(2π) = 2 where we care about finding a solution over the interval [0,27]. Suppose we attempt to numerically approximate the solution by dividing this interval into M equal-width panels, and using second-order symmetric approximations to the second derivative d²u u(x+h)-2u(x) + u(x = h) h² dx² for each of the interior grid points. In class, we derived the resulting linear system and performed an error analysis w.r.t. the mesh weighted l2 norm M e2.h = hΣ(u(xj) — uj)² j=0 where the xjs are the grid points and u; is our approximation to the true solution at xj, u(2j). Suppose we write a program to solve the resulting linear system for the numerical iter- ates, and we obtain the following table of mesh weighted errors: Panels Error 8 0.04169 16 0.02083 32 0.01041 64 0.00520 Is our program working correctly? Why or why not?