CAAM 552 · Foundations of finite element methods

Spring 2017 · Rice University


PROBLEM SETS  //  PIAZZA  //  PAPER PRESENTATIONS  // 

CLASS: 2:30PM - 3:45PM Tuesday/Thursday Abercrombie Lab A121
INSTRUCTOR: Jesse Chan (jesse.chan@rice.edu)
Duncan Hall 3023, (713) 348-6113
OFFICE HOURS: Wednesday 1-2pm or by appointment.
GRADING: 70% problem sets, 30% paper reviews
TEXTS: The Mathematical Theory of Finite Element Methods by Susanne Brenner and L. Ridgeway Scott
Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation by Beatrice Riviere
An analysis of the finite element method by Gilbert Strang and George Fix
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications by Jan Hesthaven and Tim Warburton
Mathematical Aspects of Discontinuous Galerkin Methods by Daniele Di Pietro and Alexandre Ern
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SYLLABUS: pdf
LECTURES: Lecture 1: Introduction, one-dimensional formulations (Brenner and Scott, Chapter 1)
Lecture 2: weak derivatives, Sobolev spaces (Brenner and Scott, Chapter 2)
Lecture 3: trace theorems in Sobolev spaces (Brenner and Scott, Chapter 2)
Lecture 4: symmetric variational formulations, the Lax-Milgram theorem (Brenner and Scott, Chapter 2)
Lecture 5: non-symmetric variational formulations, generalized Lax-Milgram theorem (Brenner and Scott, Chapter 2)
Lecture 6: coercive variational formulations, Poisson's equation (Brenner and Scott, Chapter 5)
Lecture 7: Poisson's equation, pure Neumann boundaries, meshes (Brenner and Scott, Chapter 5, Chapter 3)
Lecture 8: Finite elements: definition and interpolants (Brenner and Scott, Chapter 3)
Lecture 9: Interpolation estimates: approximation results for averaged Taylor polynomials (Brenner and Scott, Chapter 4)
Lecture 10: Interpolation estimates (Brenner and Scott, Chapter 4)
Lecture 11: Finite element error estimates (Brenner and Scott, Chapter 5)
Lecture 12: Finite element error estimates for parabolic problems (Strang and Fix)
Lecture 13: Finite element error estimates for the 2nd order wave equation (Strang and Fix)
Lecture 14: Non-conforming methods (Brenner and Scott)
Lecture 15: Non-conforming methods: Crouzeix-Raviart, DG methods for Poisson's equation (Brenner and Scott)
Lecture 16: NIPDG methods for Poisson: coercivity and continuity (Brenner and Scott)
Lecture 17: SIPDG methods for Poisson: coercivity and continuity (Brenner and Scott)
Lecture 18: DG methods for advection-reaction: central fluxes (Di Pietro and Ern)
Lecture 19: Central flux DG for advection-reaction: coercivity, continuity, and convergence (Di Pietro and Ern)
Lecture 20: Upwind flux DG for advection-reaction: coercivity, continuity(Di Pietro and Ern)
Lecture 21: Upwind flux DG for advection-reaction: convergence (Di Pietro and Ern)
Lecture 22: Second order DG methods: SIPG in mixed formulation
Lecture 23: Second order DG methods: LDG and upwinding, variable contrast diffusivity
Lecture 24: Time-dependent hyperbolic systems: semi-discrete error analysis
Lecture 25: Time-dependent hyperbolic systems: discrete error analysis


Any student with a disability requiring accommodation in this course is encouraged
to contact the instructor during the first week of class, and also to contact
Disability Support Services in the Ley Student Center.