� � � � � � � �� � � � � � � � 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(mΔx) at points x = mΔx [for Δx = L/(M + 1)] as A = −DT D/Δx2, where ⎞⎛ 1⎞⎛ −1 1 −1 1 DDirichlet = ⎜⎜⎜⎜⎜⎝ ⎟⎟⎟⎟⎟⎠ , DNeumann = −1 1⎜⎜⎜⎝ ⎟⎟⎟⎠ . .. .. . . .. .. . −1 1 −1 1 −1 when the boundary conditions are Dirichlet u(0) = u(L) = 0 [D an (M +1)×M matrix] and Neumann u�(0) = u�(L) = 0 [D an (M − 1) × M matrix], respectively. (a) Write down a new D matrix that implements the boundary conditions: u(0)+u�(0) = 0, u(L) = 0. For simplicity, apply the left boundary condition at Δx/2 rather than at 0, using u(Δx/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 u��(x) = f(x) approximately via A = −DT D/Δx2, how nfast would you expect the errors to vanish as Δx 0? [i.e. errors proportional to Δx 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 ˆ2 ∂ ∂u ∂ ∂u Au = � u + ∂x c(x, y) ∂y + ∂y c(x, y) ∂x for some real-valued function c(x, y). Let Bˆ= c , i.e. ˆc(x, y) ∂u A. ∂ ∂ Bu = ∂ , the second term in ˆ∂x ∂y ∂x ∂y � Lx � Ly 0 0 dx dy u(x, y) v(x, y) [use(a) Find Bˆ∗ =_____________ under the inner product �u, v� = uv = Ω real-valued functions for simplicity]. Hence conclude that Aˆ∗ =________. ∂u c(x, y) will ˆhave solutions 0 as t → ∞ for any initial condition Au = ∂t u(x, y, t) that(b) Under what conditions on → 1 cˆu(x, y, 0)? Hint: consider �u, Au� for u = 0, and note that the eigenvalues of the 2 × 2 matrix are c 1 1 ± c. (From class: �u, �2u� = − Ω �u · �u.) Problem 3: Thinking Green (20 points) Consider the operator Aˆ= −c(x)�2 in some 2d region Ω ⊆ R2 with Dirichlet boundaries (udΩ = 0), where c(x) > 0. Suppose the eigenfunctions of Aˆare un(x) with eigenvalues λn [that is, ˆ= λnun] for n|=Aun 1, 2, . . ., numbered in order λ1 < λ2 < λ3 < . Let G(x, x�) be the Green’s function of Aˆ.· · · (a) If f(x) = n αnun(x) for some coefficients αn =_________________ (expression in terms of f and un), then Ω G(x, x�)f(x�)d2x� =__________________ (in terms of αn and un). (b) The maximum possible value of1 u(x)G(x, x�)u(x�) d2x d2x�Ω Ω c(x) Ω |uc((xx����))|2 d2x�� , for any possible u(x), is _____________________ (in terms of quantities mentioned above). [Hint: min–max.] 1MIT OpenCourseWare http://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and Numerics Fall 2010 For information about citing these materials or our Terms of Use, visit:
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