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Find the solution to the Poisson equation d² dx²u(x) = x²(1 − x) in the unit interval [0, 1] using homogeneous Dirichlet conditions on the two boundaries, u(0) = u(1) = 0. Use a uniform a grid of size n = 100. Solve the system using (i) Jacobi, and (ii) Gauss-Seidel iterations. Plot (semilogy) the residual as a function of iteration number. Is this behavior expected and why?

Find the solution to the Poisson equation d² dx²u(x) = x²(1 − x) in the unit interval [0, 1] using homogeneous Dirichlet conditions on the two boundaries, u(0) = u(1) = 0. Use a uniform a grid of size n = 100. Solve the system using (i) Jacobi, and (ii) Gauss-Seidel iterations. Plot (semilogy) the residual as a function of iteration number. Is this behavior expected and why?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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PROBLEM 1 Find the solution to the Poisson equation d² dx2 u(x) = x²(1 − x) in the unit interval [0, 1] using homogeneous Dirichlet conditions on the two boundaries, u(0) = u(1) = 0. Use a uniform a grid of size n = 100. Solve the system using (i) Jacobi, and (ii) Gauss-Seidel iterations. Plot (semilogy) the residual as a function of iteration number. Is this behavior expected and why?
Transcribed Image Text:PROBLEM 1 Find the solution to the Poisson equation d² dx2 u(x) = x²(1 − x) in the unit interval [0, 1] using homogeneous Dirichlet conditions on the two boundaries, u(0) = u(1) = 0. Use a uniform a grid of size n = 100. Solve the system using (i) Jacobi, and (ii) Gauss-Seidel iterations. Plot (semilogy) the residual as a function of iteration number. Is this behavior expected and why?
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