MIT Dept. of MathematicsBenjamin Seibold18.086 spring 2007Exercise Sheet 4Out Mon 04/02/07Due Wed 04/18/07We would like to solve a Poisson problem on the 2d geometry given in Figure 1, andthe 3d geometry given in Figures 2 and 3.Let the interior of the domains be denoted Ω, and the boundary denoted Γ. In the2d geometry, the boundary consists of lines, in the 3d geometry, the boundary iscomposed of f aces. The Poisson problem to be solved is−∆u = 1 in Ωu = f on ΓD∂u∂n=∂f∂non ΓNLet the f unction f , which defines the boundary conditions, be• f(x, y) = x3y − xy3−12x2in the 2d case,• f(x, y, z) = x3yz + xy3z − 2xyz3−12x2in the 3d case.Let Dirichlet boundary conditions be specified where boundary lines are thin andfaces are light, and Neumann boundary conditions be specified where boundary linesare thick and faces are dark.Fig 1: Tetris geometry(2d)Fig 2: Blockout geometry(3d) f rom the frontFig 3: Blockout geometry(3d) from behindExercise 8 Write a matlab program which takes a grid resolution parameter as inputand sets up a linear system discretizing the above 2d and 3d Poisson problems.Exercise 9 For both the 2d and the 3d problem, choose a small resolution, a mediumresolution and a high resolution, and (try to) solve the linear systems by1. elimination with reordering2. Jacobi iteration3. a simple multigrid method (V-cycle)4. conjugate gradientsCompare the different approaches with respect to accuracy and run times.Remarks:• For elimination, matlab functions can be used. The other three methods haveto be imple mented by hand (in a simple version).• In all cases, the numerical error due to solving the linear system shall not besignificantly larger than the error due to the discretization of the geometry.• Some solution methods might fail for very large resolutions. T he medium reso-lution shall be chosen, such that all above solution methods succeed.• The high resolution shall drive matlab to its limits. Fame and glory for thestudent who can manage the largest res olution in his/her