Consider the boundary value problem u u x, for 0 x , with u(0) 0, u(1) 0 (a) For what values of a does the system have a unique solution (b) For which values of can you find a minimization principle that characterizes the solution Is the solution unique for all such values of A (c) Using n equally ...

a There is a unique solution provided n 2 for n 1 2 3 namely When n 2 the boundary value problem has no solution b The minimization principle dx over ...

Consider the boundary value problem - u+ u = x, for 0 < x < , with

Question:

Consider the boundary value problem - u"+ λu = x, for 0 < x < π, with u(0) = 0, u(1) =0.
(a) For what values of a does the system have a unique solution?
(b) For which values of λ can you find a minimization principle that characterizes the solution? Is the solution unique for all such values of A?
(c) Using n equally spaced mesh points, write down the finite element equations for approximating the solution to the boundary value problem. Note: Although the finite element construction is only supposed to work when there is a minimization principle, we will consider the resulting linear algebraic system for any value of λ.
(d) For which values of A does the finite element system have a unique solution? Hint: Use Exercise 8.2.48. How do these values compare to those in part (a)?
(e) Select a value of A for which the solution can be characterized by a minimization principle and verify that the finite element approximation with n = 10 approximates the exact solution.
(f) Experiment with other values of A. Does your finite element solution give a good approximation to the exact solution, when it exists? What