lectures

Lecture Notes

Here is a list of lecture notes and projects used mainly for the course Math 226: Computational PDEs in UC Irvine. I try to strike a good balance between mathematical theory and programming skill. Welcome to send me your comments (e.g. typos, mistakes, notation inconsistence, suggestion, and even complains) on the lecture notes.

Calculus of Variation

1. Introduction
2. Classic theory
3. Existence of global minimums
4. Hamilton Jacobi theory
5. Constraint Variation Problems
6. Noether's theory

Basic Numerical Methods

1. Introduction
2. Finite Difference Methods
3. Programming of Finite Difference Methods
4. Project: Finite Difference Method.
5. Finite Volume Methods

Numerical Analysis

1. Sobolev Spaces
2. Inf-sup Conditions
3. Unified Error Analysis

Virtual Element Methods

1. Programming of Linear Virtual Element Methods

Finite Element Methods

1. Finite Element Methods
2. Programming of Finite Element Methods
3. Project: Finite Element Method.

Adaptive Finite Element Methods

1. Introduction to Adaptive Finite Element Methods
2. Convergence Theory of Adaptive Finite Element Methods
3. Data Structure for Triangulations

Iterative Methods

1. Classical Iterative Methods
2. Conjugate Gradient Methods
3. Subspace Correction and Auxiliary Space Methods

Multigrid Methods

1. Introduction to Multigrid Methods
2. Programming of Multigrid Methods
3. Project: Multigrid Methods.
4. Recrusive Proofs of Multigrid Methods

Multilevel Algorithms

1. Quick sort and Merge sort
2. FFT (Fast Fourier Transform)
3. Fast Multipole Methods
4. Project: Fast Multipole Methods.

Nonlinear Equations

1. Nonlinear Elliptic PDE
2. Project: Nolinear Poisson-Boltzmann Equations.

Parabolic Equations

1. Finite Differnce Methods for Parabolic Equations
2. Finite Element Methods for Parabolic Equations
3. Project: Heat Equations.

Numerical methods in CFD

1. Brief introduction to Navier-Stokes equation
2. Finite element methods for Stokes equations
3. Project: Finite element methods for Stokes Equations.
4. MAC scheme for Stokes equations
5. Programming of MAC Scheme for Stokes Equations
6. Project: MAC scheme for Stokes Equations.
7. Fast solvers for Stokes equations

Finite Element Methods for Linear Elasticity

1. Introduction to Linear Elasticity
2. Variational Formulation of Linear Elasticity
3. Tensor Calculus
4. Finite Element Methods for Linear Elasticity
5. Project: ERobust Finite Element for Linear Elasticity

Numerical methods in Electromagnetism

1. Brief Introduction to Maxwell's Equations
2. Finite Element Methods for Maxwell's Equations
3. Multigrid for H(curl) and H(div) Problems
4. Project: Edge Element for Maxwell Equations