|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. Lp-stability. Conditioning.|
|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.|
|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%.