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18.303 Midterm, Fall 2010 Each problem has equal weight. You have 55 minutes. Problem 1: Finite differences (20 points) From class and homework, the d2 /dx2 operator on [0, L] can be discretized into values um ≈ u(m∆x) at points x = m∆x [for ∆x = L/(M + 1)] as A = −DT D/∆x2 , where 1 −1 1 −1 1 −1 1 .. .. DNeumann = DDirichlet = , . . . . .. .. −1 1 −1 1 −1 when the boundary conditions are Dirichlet u(0) = u(L) = 0 [D an (M +1)×M matrix] and Neumann u0 (0) = u0 (L) = 0 [D an (M − 1) × M matrix], respectively. (a) Write down a new D matrix that implements the boundary conditions: u(0)+u0 (0) = 0, u(L) = 0. For simplicity, apply the left boundary condition at ∆x/2 rather than at 0, using u(∆x/2) ≈ (u0 + u1 )/2. Be sure to indicate how many rows and columns your D matrix has, and for what m values you have degrees of freedom um . (b) If you use your D matrix from the previous part to solve u00 (x) = f (x) approximately via A = −DT D/∆x2 , how fast would you expect the errors to vanish as ∆x → 0? [i.e. errors proportional to ∆xn for what power n?] Problem 2: Adjoints and stuff (20 points) Let Ω ⊆ R2 be the rectangular 2d region x ∈ [0, Lx ], y ∈ [0, Ly ], with Dirichlet boundaries u|dΩ = 0. Consider the operator ∂ ∂u ∂ ∂u 2 Âu = ∇ u + c(x, y) + c(x, y) ∂x ∂y ∂y ∂x h i ∂ ∂ ∂ for some real-valued function c(x, y). Let B̂ = ∂x c(x, y) ∂u c ∂y , i.e. B̂u = ∂x ∂y , the second term in Â. R RL RL (a) Find B̂ ∗ =_____________ under the inner product hu, vi = Ω uv = 0 x 0 y dx dy u(x, y) v(x, y) [use real-valued functions for simplicity]. Hence conclude that Â∗ =________. ∂u ∂t have solutions u(x, y, t) that → 0 as t → ∞ for any initial condition 1 c u(x, y, 0)? Hint: consider hu, Âui for u 6= 0, and note that the eigenvalues of the 2 × 2 matrix are c 1 R 2 1 ± c. (From class: hu, ∇ ui = − Ω ∇u · ∇u.) (b) Under what conditions on c(x, y) will Âu = Problem 3: Thinking Green (20 points) Consider the operator Â = −c(x)∇2 in some 2d region Ω ⊆ R2 with Dirichlet boundaries (u|dΩ = 0), where c(x) > 0. Suppose the eigenfunctions of Â are un (x) with eigenvalues λn [that is, Âun = λn un ] for n = 1, 2, . . ., numbered in order λ1 < λ2 < λ3 < · · · . Let G(x, x0 ) be the Green’s function of Â. P (a) If f (x) = R n αn un (x) for some coefficients αn =_________________ (expression in terms of f and un ), then Ω G(x, x0 )f (x0 )d2 x0 =__________________ (in terms of αn and un ). (b) The maximum possible value of 1 u(x)G(x, x0 )u(x0 ) d2 x d2 x0 Ω Ω c(x) , R |u(x00 )|2 d2 x00 Ω c(x00 ) R R for any possible u(x), is _____________________ (in terms of quantities mentioned above). [Hint: min–max.] 1