MATH 350 Final, Fall 2007

AMERICAN UNIVERSITY OF BEIRUT Faculty of Arts and Sciences Mathematics Department MATH-CMPS 350 FINAL EXAMINATION-Make-up FALL 2007-2008 Two hours STUDENT NAME ID NUMBER Problem Out of Grade 1 2 3 4 5 6 TOTAL 1 Let Ω ⊂ R2 be an open connected domain with Γ = ∂Ω = ΓD ∪ ΓN, ΓD ∩ΓN = Φ (empty set). Let a, b f be continuous functions (C(Ω)), such that: a(x, y) ≥ a0 > 0, b(x, y) ≥ 0, ∀(x, y) ∈ Ω. Consider the problem of finding u : Ω → R, that verifies: (P) � −div(a(x, y)∇u) + b(x, y)u = f(x, y) in Ω u = 0 on Γ 1. State Green’s formula and derive the variational (weak) form of (P) in the form: (P’) u ∈ V : A(u, v) = F(v), ∀v ∈ V. Define the Sobolev space H1(Ω) and specify the bilinear form A(., .), the functional F(.), and the subspace V . 2 2. Prove existence and uniqueness of the solution to (P’). 3 3. Consider the one-dimensional version corresponding to the problem of question 1, i.e. let: Ω = (−1, 1); T = {v ∈ H1(Ω)|v(−) = v(1) = 0}; A(u, v) = � Ω (a(x)u ′v ′ + b(x)uv)dx; F(v) = � Ω (f(x)v)dx, where a(x) ≥ a0 > 0 and b(x) ≥ 0. Consider the variational problem: u ∈ T : A(u, v) = F(v), ∀v ∈ T. (1) Assume u ∈ H2 and let SN = span{Qn|n = 0, ..., N}, N > 2 be the orthonormal set of Legendre polynomials in L2(Ω). Give an esti- mate on ||u ′ − ΠN−1(u′)||, where ΠN−1(u′) is the L2 projection of u ′ on SN−1. 4 4. By studying the inner product < u ′, Q0 >, show that: TN = span{ϕn|n = 1, ..., N − 1}, with: ϕn = Pn+1(x) − Pn−1(x), n = 1, 2, ..., N − 1. is a suitable subspace of polynomials of degree n for a Galerkin approxi- mation to (1). Give an estimate on ||u − rN(u)||1, where rN(u)(x) = � x −1 ΠN−1(u ′). 5 5. Let uN be the Galerkine spectral approximation uN ∈ TN uN(x) = ΣN n=1cnϕn(x). Give the Galerkin formulation based on (1), uN ∈ TN : (2) (a) Give the system of linear equations: Ac = F; c ∈ RN, A ∈ RN,N, F ∈ RN. (3) equivalent to (2). Specify the matrix A the vectors F and c. Give (with proof) the properties of A. 6 (b) Prove the estimation: ||u − uN||1 ≤ c min v∈TN ||u − v||1 7 6. Let A be an n×n inverible matrix. Let x0 ∈ Rn and b ∈ Rn, r0 = b−Ax0, and x the unique solution to Ax = b. Prove that: x ∈ x0 + span{r0, Ar0, ..., An−1r0}. 8