Math 540: Numerical Computation
Math 540 is a graduate level introduction to the essentials of scientific computation. The course is based on the variational approach which offers a number of advantages. In particular, a significant amount of time is spent on the Finite Element Method which is variational in nature.
The course is designed with Engineers in mind however, being an introductory course, it is also appropriate for students from all fields including Mathematics and Computer Science. In order to succeed in the course you must be comfortable with the fundamentals of Linear Algebra and Multivariate Calculus.
This course is all about implementation! We introduce a number of tools and adhere to the philosophy different problems call for different tools. Where appropriate we use Excel, for more complicated problems - Matlab.
| I. Rayleigh-Ritz | ||
|---|---|---|
Shape of hanging string. | Why are droplets spherical? | Droplets on substrates |
| II. Calculus Review | |||
|---|---|---|---|
How to find minima of functions | Mutlivariable functions | Constrained minimization | Lagrange multipliers |
| III. Linear Algebra Review | ||
|---|---|---|
Quadratic forms | Positive definite quadratic forms | Quadratic form minimization |
| IV. Finite Elements in 1D | ||
|---|---|---|
Introduction. Laplace's equation | The finite element basis | The stiffness matrix |
| V. Finite elements in 2D and 3D | ||
|---|---|---|
Triangular meshes | The FEM basis | Tetrahedral meshes |
| VI. Calculus of variations | |||
|---|---|---|---|
Variational formulation | Derivation of Laplace's equation | Neumann or natural BC | |
| VII. Quadratic finite elements | ||
|---|---|---|
Quadratic elements | The FEM basis | Convergence implications |
| VIII. Problems from Physics and Engineering | ||
|---|---|---|
The Poisson equation | Equations of elasticity | The heat equation |
| IX. Numerical Linear Algebra | ||
|---|---|---|
Conjugate gradient method | Power iteration | Overview of other methods |
| X. Additional Topics | |
|---|---|
| Singular finite elements | |