Iterative methods to solve a linear system

Institute of Scientific ComputingTechnical University BraunschweigProf. Hermann G. MatthiesDr. Alexander LitvinenkoWinter Term 2012/201313.11.2012Introduction to PDEs and Numerical Methods:Assignment 4: Iterative Linear SolversExercise 1: Iterative Linear SolversDownload the matlab library for iterative solvers from the semesterapparat (Prof. H. G. Matthies)http://www.biblio.tu-bs.de/semapp/Start programs elltrit1.m for 1D example and elltrit2.m for 2D exampleStart different iterative methods to observe convergence.Look inside these programs and find how the convergence is evaluated.Exercise 2: Iterative Linear SolversConsider the stationary heat equation(40 points)−β ∂2 u(x)∂x 2 = fon the domain [0, 100] with β = 2.0 and boundary conditions u(0) = 0 and u(100) = 1. Discretisingthe equation by using finite differences leads toAu = f.In this exercise no explicit source/sink terms should be introduced. Remark: the analytical solutionis u(x) = 1100 x.(a)The SOR method can be derived by choosing M := (D + ωL) in the matrix splitting methodMu n+1 = Mu n + ω(f − Au n ),where A = D + L + U with diagonal matrix D, strictly lower triangular matrix L and strictly uppertriangular matrix U of A.Write a MATLAB function realising the SOR method. The signature of the function shall be[u, err, iter] = my sor( A, f, u0, tol, max iter )with matrix A, right hand side f, start vector u0. Use the discrete norm of the residual ||Au − f|| asthe error of the method with||a|| = √ 1 nn∑a 2 i , a ∈ Rn .i=1Stopping criterion: When the error is smaller than tol or the number of iterations larger thanmax iter, the method shall stop and return the actual solution u, the achieved error err and the

equired number of iterations iter.(15 points)(b) Write a MATLAB code which uses your SOR implementation to solve the linear system, specifiedat the beginning of this exercise.(10 points)(c)Extend your MATLAB code of (b), so that you can plot the convergence of SOR with respectto the given system. For it, plot the relative error||u − u it ||||u||over the number of iterations, where u is the analytical solution given by the mentioned functionand u it the approximated one got from the iterative method. For each number of discrete points –50, 100, 200 – plot the relative error over the given number of iterations – 100, 200, . . . , 1000 – andchoose for ω the values 1.0, 1.5 and 1.9. Put all nine graphs into one figure. To make the figureunderstandable use different line-types, plot symbols and colors, Use also the MATLAB commandlegend to name the graphs.(15 points)