Übungen zu Theorie und Numerik partieller Differentialgleichungen

Universität Konstanz WS 10/11 Fachbereich Mathematik und Statistik

S. Volkwein, O. Lass, R. Mancini

Übungen zu Theorie und Numerik partieller Differentialgleichungen

http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/

Submission: 17.01.2011, 11:00 o’clock Codes by E-Mail and Reports in Box 18

Program 2 (8 Points)

Let Ω = (0,1)×(0,1) and h = _M¹ with M ∈ N. Solve numerically the nonlinear elliptic problem

−∆u(x, y) +f(x, y, u(x, y)) = 0, (x, y)∈Ω

u(x, y) = g(x, y), (x, y)∈∂Ω (1)

with the Newton method. For the discretization of the Laplace operator use the classical finite difference scheme (i.e. five-point-stencil). Use the lexicographical ordering of the grid points in Ω. Use the following functions f and g:

f(x, y, u(x, y)) =e^u(x,y) and g(x, y) =

½ x+y, if |x+y| ≤1, 2−x−y, if |x+y|>1.

a) Implement the Newton method for solving problem (1). As a stopping criteria for the Newton method use the norm of the residual to be less than10⁻⁶. For the initial guess choose u⁰ = (0.5, . . . ,0.5)^> ∈R^(M−1)2. For M we choose 100. For solving the linear system use the \-operator in Matlab.

b) Implement the Conjugate-Gradient (CG) method, Algorithm 1, and use it to replace the \-operator to solve the linear system arising from the Newton method in step a). For the values x⁰ and ² in Algorithm 1 use (0, . . . ,0)^> ∈ R^(M−1)

2 and 10⁻⁶, respectively.

Visualize the numerical solutionu(x, y)and do not forget to lable the plots. In the written report derive the Newton step for problem (1). Verify that the Jacobian obtained in the Newton method is positive definite.

Algorithm 1 (CG-Algorithm)

1: Given: A∈R^n×n symmetric positive definite,b ∈Rⁿ, x⁰ ∈Rⁿ and ² >0

2: Set: k = 0, r⁰ =b−Ax⁰ and p⁰ =r⁰

3:

4: while kr^kk> ² and k < ndo

5: α_k= (p^k)^Tr^k (p^k)^TAp^k

6: x^k+1 =x^k+α_kp^k

7: r^k+1 =r^k−α_kAp^k

8: β_k= (Ap^k)^Tr^k+1 (Ap^k)^Tp^k

9: p^k+1 =r^k+1−β_kp^k

10: k=k+ 1

11: end while