## Mathematics 16:642:575 Numerical Solution of Partial Differential Equations

### Schedule

The course is usually offered every two years during the Spring semester.- Class meeting dates: Please visit the University's academic calendar.
- Schedule and Instructor: Please visit the University's schedule of classes for the instructor, time, and room.
- Instructor and Teaching Assistant Office Hours: Please visit the Mathematical Finance program's office hour schedule.

### Course Abstract

In this course, we study finite difference, finite element, and finite volume methods for the numerical solution of elliptic, parabolic, and hyperbolic partial differential equations. The course will concentrate on the key ideas underlying the derivation of numerical schemes and a study of their stability and accuracy. Students will have the opportunity to gain computational experience with numerical methods with a minimal of programming by the use of Matlab's PDE Toolbox software. Since there are many sophisticated computer packages available for solving partial differential equations, one might think that a thorough understanding of the numerical methods employed is no longer necessary. A striking example of why naive use of such codes can lead to disastrous results is the sinking of the Sleipner A offshore oil platform in Norway in 1991, resulting in an economic loss of about $700 million. The post accident investigation traced the problem to inaccurate finite element approximation of the linear elastic model of the structure (using the popular finite element program NASTRAN). The shear stresses were underestimated by 47%, leading to insufficient design. More careful finite element analysis, made after the accident, predicted that failure would occur with this design at a depth of 62m, which matches well with the actual occurrence at 65m.### Pre-requisites

At least one of Numerical Analysis I (16:642:573) or Numerical Analysis II (16:642:574).### Primary Textbooks

Dietrich Braess, Finite Elements: Theory, fast solvers, and applications in solid mechanics, 3rd ed., Cambridge University, 2007.

Note: Most of lectures will be based on hand outs prepared by the instructor. Students may have one of the aforementioned textbooks depending on their preference.### Grading

Please contact the instructor.### Class Policies

Please see the MSMF common class policies.### Assignments

Homework assignments in the course consist of both theoretical and computational work. For the computational component, the students should use a language/environment that possesses high level data types so that the students spend more time working with algorithms and not worrying about the details of writing computer code. MATLAB is a good choice. Fortran 77/90/95 and C++ with appropriate class libraries can also be used. There will be one assignment for each 3-4 class periods. Since we intend to hand out solutions, late homework assignments pose a problem. Students with exceptional circumstances may be granted short extensions. Please contact the instructor as soon as a problem arises.### Previous Instructor Course Websites

2010 Young-Ju Lee2008 Richard Falk

### Weekly Lecturing Agenda and Readings

The lecture schedule below is a sample; actual content may vary depending on the instructor.

Lecture | Topics |
---|---|

1 | Finite Difference Methods for Elliptic Problems |

2 | Stability and Error Estimates |

3 | Extensions of the Method |

4 | Finite Element Method for Elliptic Equations - Introduction |

5 | Finite Element Method for Elliptic Equation |

6 | Construction of finite element subspaces |

7 | Affine families of finite elements |

8 | Error estimates for piecewise linear interpolation |

9 | Error estimates by scaling |

10 | Order of Convergence and other Finite Elements |

11 | Approximation of saddle point problems |

12 | Error estimates for the approximation of saddle point problems |

13 | Application to the mixed finite element method for Poisson's equation |

14 | Application to the stationary Stokes equations |

15 | Efficient solution of the linear systems arising from finite element discretization |

16 | Efficient solution of the linear systems arising from finite element discretization |

17 | Finite difference methods for the heat equation |

18 | Finite difference methods for the transport equation and the wave equation |

19 | Stability of difference schemes for pure IVP with periodic intial data |

20 | Stability of difference schemes -- examples |

21 | Qualitative properties of finite difference schemes |

22 | Finite element methods for parabolic problems |

23 | A finite element method for the transport equation |

24 | Approximation of hyperbolic conservation laws |