## Math 228B - Numerical Solution of Differential Equations

Instructor: J Strain

Lectures: Tuesday and Thursday 11:00-12:30, 75 Evans.

Office Hours: Tuesday 12:30-2:00 and Wednesday 1:00-2:30, 1099 Evans

GSI: John DeIonno

Office Hours: Tuesday 12:30-1:30 in LaVal's, 1:30-3:30 in 1042 Evans.

Prerequisites: Math 128A or equivalent knowledge of basic numerical analysis. Sufficient computer skills and gumption to download, compile, modify and run numerical packages written in Fortran and C.

Required Text: A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences no. 159, Springer-Verlag, New York, 2004.

Syllabus: The course will cover the theory and practice of finite element methods for partial differential and integral equations.

• Basic examples, functional analysis, interpolation, approximation.
• Galerkin methods.
• Coercive problems. Lax-Milgram lemma.
• Saddle point problems. Inf-sup conditions.
• First-order PDEs. Flow problems. Discontinuous Galerkin.
• Time-dependent problems. Space-time finite elements.
• Implementation. Data structures. Triangulation. Quadrature. Fast solvers. Error estimation.
• Grading: Grades will be based on weekly problem sets, some individualized.

Lecture Notes:

• Week 01: PDF PS TeX
• Week 02: PDF PS TeX
• Week 03: PDF PS TeX
• Problem Sets:

• Problem Set 01 (due Thursday February 5): PDF PS TeX
• Solutions: PDF
• Problem Set 02 (due Thursday February 19): PDF PS TeX
• Solutions: PDF
• Problem Set 03 (due Thursday March 18): Solve exercises 3.2, 3.4 and 8.10 of Ern-Guermond.
• Solutions: PDF
• Problem Set 04 (due Thursday April 2): Solve exercises 3.5 and 3.7 of Ern-Guermond.
• Solutions: PDF
• Problem Set 05 (due Thursday April 30): Solve exercises 9.1, 9.2, 9.3, 9.4 and 9.11 of Ern-Guermond.
• Problem Set 06 (due Thursday May 7): Solve exercises 10.8, 10.10, 4.4 and 4.8 of Ern-Guermond.