Programming Poisson's Equation: Error Analysis and Solution Visualization - Prof. Yanqiu W, Assignments of Mathematics

Download Programming Poisson's Equation: Error Analysis and Solution Visualization - Prof. Yanqiu W and more Assignments Mathematics in PDF only on Docsity! Programming assignment 4 1. Solve the Poisson’s equation in Ω = (0, 1) × (0, 1): { −∆u = 2π2 sin(πx) sin(πy) in Ω, u = 0 on ∂Ω, by the standard five-point finite difference scheme on a uniform M × M grid. The step size is h = 1/M . The exact solution for this problem is u = sin(πx) sin(πy). (a) Compute the maximum norm of error ‖u − v‖∞ on grid points, where v is the finite difference solution. Report the error for 16 × 16, 32 × 32, 64 × 64 and 128 × 128 grids. Does your result agree with the error estimate ‖u − v‖∞ = O(h 2). (b) Plot the numerical solution on a 20 × 20 grid. Note: • You may use the Matlab build-in function “delsq” to generate the stiffness matrix. However, you will need to read the help file and make sure you use it correctly. • To solve a linear system Ax = f , where x, and f are n-dim column vectors, you can use the Matlab command “x = A\ f”. • A useful Matlab command is “reshape”, which returns a matrix whose elements are taken column-wise from a given vector or matrix. For example, let column vector x = [1, 2, 3, 4]t. “y=reshape(x,2,2)” will return a 2 × 2 matrix y = ( 1 2 3 4 ) . Then you can use command “surf(y)” to plot the matrix as a surface. 1