Date | Topics |
T 1/7 | Hilbert space method. Sobolev spaces. Strong derivative. Weak formulation. |
R 1/9 | Weak derivative. Poisson problem. Friedrichs inequality. |
T 1/14 | Boundary conditions. Variational and operator formulations. Ritz-Galerkin framework. |
R 1/16 | Céa's lemma. Fourier-Galerkin methods. |
T 1/21 | Finite elements in 1D. |
R 1/23 | Finite elements in 2D. Averaged Taylor polynomials. |
T 1/28 | Error of averaged Taylor polynomials. Truncated Riesz potential. |
R 1/30 | Bramble-Hilbert lemma. Nodal interpolation for Lagrange finite elements. |
T 2/4 | Mesh refinement. Discontinuous elements. Quasi-interpolation. |
R 2/6 | Inverse estimates. L^{p}-stability. Conditioning. |
T 2/11 | K-functionals. |
R 2/13 | Real interpolation. Interpolation of Lebesgue spaces. |
T 2/18 | Interpolation of Sobolev spaces. |
R 2/20 | Approximation spaces. |
T 2/25 | Abstract approximation theory. |
R 2/27 | Application to finite elements. Zygmund spaces. |
3/2–3/6 | Study break |
T 3/10 | Besov spaces. Inf-sup conditions. |
R 3/12 | Saddle point problems |
T 3/17 | Class cancelled |
R 3/19 | Class cancelled |
T 3/24 | Class cancelled |
R 3/26 | Class cancelled |
T 3/31 | Petrov-Galerkin methods. Fortin operators. |
R 4/2 | Finite elements for the Stokes problem. Raviart-Thomas elements. |
T 4/7 | Variational crimes. Numerical quadrature. |
R 4/9 | Nitsche's boundary penalty method. |
Instructor: Dr. Gantumur Tsogtgerel
Prerequisite: MATH 475 and MATH 387 or permission of the instructor
Note: If you plan to take this course without taking MATH 578, please consult with the instructor.
Topics: The main focus of the course is going to be on mathematical analysis of finite element methods. If time permits, topics on adaptivity will be included.
Calendar description: Numerical solution of initial and boundary value problems in science and engineering: ordinary differential equations; partial differential equations of elliptic, parabolic and hyperbolic type. Topics include Runge Kutta and linear multistep methods, adaptivity, finite elements, finite differences, finite volumes, spectral methods.
Homework: Both analytical and computational. Assigned and graded roughly every two weeks.
Course project: The course project consists of the student studying an advanced topic, implementing the relevant algorithms, experimenting, writing a report, and giving a presentation.
Grading: Homework assignments 50% + Course project 50%.