3072 Math Finite Element Methods
Homework 3, Due February 21, 2007 1. Consider the boundary value problem A u = f in u = gD on D u + A u n = gR on R , (1) (2) (3)
where = D R , |D | > 0, 0 (x) 1 , and A(x) is a d d symmetric matrix satisfying 0 T T A(x) 1 T a) Derive the weak formulation of (1)(3). b) Prove that there exists a unique weak solution. c) Prove that if the weak solution is in H 2 (), then it is the solution to (1)(3). d) Give the minimization formulation for (1)(3) and show that it is equivalent to the weak formulation. 2. Consider the linear operator A : V V dened in the proof of the Lax-Milgram Theorem: for V , A satises (A, v) = a(, v) v V. Prove that Range(A) = V . Follow the steps a) prove that Range(A) is a subspace b) prove that Range(A) is a closed subspace c) employ the Projection Theorem to conclude that Range(A) = V 3. a) Derive the weak formulation of the elliptic BVP (a(x)u (x)) + b(x)u (x) + c(x)u(x) = f (x), u(0) = u(1) = 0, 1 0 < x < 1, (4) (5) Rd , x .
where 0 < a0 a(x) < a1 , 0 < a0 c(x) c1 < , |b(x)| a0 for all x . b) Prove that the weak formulation has a unique solution. 4. Computational problem. Modify your code from hw1 to solve the BVP (a(x)u (x)) + b(x)u (x) = f (x), u(0) = , u(1) = , 0 < x < 1, (6) (7)
a) Run your code for the problem with a(x) = x2 + 1, b(x) = 2x + 1, and a true solution u(x) = ex + sin(3x). Plug the true solution into the equation to obtain f (x). Take = 1, 100. b) Run your code for a(x) = .01, b(x) = 1, f (x) = 0, = 0, and = 1. The true solution in this case is u(x) = 1 eRx , 1 eR
where R = b/a (R is called the Peclet number). For all cases above: Test the convergence rates u uh
1
Chr ,
u uh Chp ,
i.e, determine r and p . To do this, run the code on a sequence of uniform grids with 1/h = 20, 40, 80, 160. Discuss your results. Plot the computed solution for 1/h = 20, 80. c) Run your code for a(x) = .001, b(x) = x 1/2, f (x) = 0, = 0, and = 1. Plot the computed solution for 1/h = 20, 40, 80, 160. Discuss your results. Submit your code.
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