Kf1 - Applied Mathematics, TMA 225 - 2010

Teachers Lecturer and examiner: Nils E M Svanstedt, telefon: 772 5346, nilss@chalmers.se Matematiska Vetenskaper Teaching assistant: Hermann Douanla douanla@chalmers.se Course content Mathematical modeling in 1D and 2D of physical processes including reaction, production and transport processes like diffusion and convection. The models are typically based on conservation laws for mass and energy and constitutive laws which are studied by means of partial differential equations (PDE). A major goal of the course is to solve these PDE numerically by the finite element method (FEM). We begin by study piecewise linear functions and their use in the approximation of given functions. We will then develop FEM using the linear approximation ideas. During the course you will work both theoretically and computationally (implementation in matlab). In particular you will develop your own FEM-solvers in Matlab. You will also learn how to use the software Puffin Body & Soul. Finally there is an introduction to function series. Lectures and computer sessions Lectures (4 h/week) and computer sessions in studio (4 h/week). Working continuously with the weekly problems is an important moment. Schedule Lectures: Monday 13-15, in KS11. Thursday 13-15, in KS101. Studio session: Tuesday 13-15, in KD1. Thursday 8-10, i KD1. Literature The Finite Element Method; Theory, Practice and Implementation, Larson and Bengzon, 2009.(LB). Calculus, Adams. Detailed program LV1: Lecture 1, Monday 15 March 13-15: Linear functions in 1d. The space of piecewise linear continuous functions in 1d. Linear interpolation. Piecewise lienear continuous interpolation. Interpolation error estimates. Proof of interpolation error estimates in maximum norm. Chapter 1.1-1.2 in LB. Lecture 2, Thursday 18 March 13-15: Introduction to L2-projection. Definition and derivation of the linear system of equations. Statement and proof of error estimates in L2-projection. Chapter 1.3 in LB. Studio 1 (och 2), Tuesday 16 March 13-15 and Thursday 18 March 8-10: Piecewise Polynomial Lab. Download the two files needed: PP.fig och PPmod.m. LV2: Lecture 3, Monday 22 March 13-15: Quadrature/numerical integration with application to solving the system for L2-projection. Studio 3, Tuesday 23 March 13-15: Review of quadrature formulas with application to numerical integration. These will immediately be used in the numerical L2-projection in the studio. Create a folder Studio3 in your cathaloge and download the files LoadVector.m and MassMatrix.m to this folder. You will be guided through the session by opening: Kvadratur. L2-projektion. Studio 4,Thursday 25 March 8-10: Continue working with Kvadratur. L2-projektion.. Lecture 4, Thursday 25 March 13-15: Finite Element Method FEM in 1D. Existence and uniqueness of FEM solution. A priori error estimate. Chapter 2.1 and 2.2 in LB. Spring Break! LV3: Lecture 5, Monday 12 april 13-15 Stationary heat equation, Robin boundary conditions. Robin boundary conditions in FEM. Chapter 2.3-2.5 in LB Studio 5, Tuesday 13 april 13-15 Open and save the m-files PoissonSolver1D.m and PoissonAssembler1D.m You are now guided through the session (in swedish!!) by clicking FEM i 1D Studio 6, Thursday 15 april 8-10: FEM for Time dependent problems. You are now guided through the session (in swedish!!) by clicking Time dependent problems. Lecture 6, Thursday 15 april 13-15: Forward and backward Euler method for numerical solution of ordinary differential ekvationes. Finite element discretization of time dependent problems. Chapter 5.0 and 5.1 in LB. LV4: Lecture 7 , Monday 19 april 13-15: Problem solving among Räkneuppgifter lv1-lv3. Studio 7, Tuesday 20 april 13-15: Continue working on Tidsberoende problem. and Project 1. Studio 8, Thursday 22 april 8-10: Finalize Project 1. Lecture 8 , Thursday 22 april 13-15: Problem solving. Examination 1. To pass two compulsory projects. Project 1 is to write a time dependent FEM solver in 1d. Run the program and demonstrate this to the teaching assistant Project 2 is described below. The projects are individual but collaboration is encouraged. 2. A written exam based on the excercises that will be assigned during the course Project 2 Written report where you set up a PDE model for a physical phenomena like convection-diffusion in 2D. Formulate a FEM problem for this PDE. Solve numerically the PDE in 2D by FEM. Especially important is the visualisation of the solutions. Problems (Räkneuppgifter): Räkneuppgifter vecka 1 Lösningar vecka 1 Räkneuppgifter vecka 2 Lösningar vecka 2 Räkneuppgifter vecka 3 Lösningar vecka 3 Extra materiel: Extra materiel om Kvadratur (1D) Extra materiel om Robin randvillkor (1D) Extra materiel om Tidsberoende problem (1D): Inga nya räkneuppgifter för vecka 4. Slutfor Inlamningsuppgift 1. Räkneuppgifter vecka 5 Lösningar vecka 5 Räkneuppgifter med lösningar vecka 6 Räkneuppgifter med lösningar vecka 7

Editor: Nils E M Svanstedt Last modified: 2010-04-14