ES 105 Winter 2012

Computational Methods for Partial Differential Equations: Class Notes

Keith D. Paulsen

WEEK 1
  • Introduction
  • Review of Finite Differences
  • Cross Derivative Difference Formulas WEEK 2
  • FD for Elliptic Equations
  • FD for General Elliptic Equations
  • Point Iterative Methods
  • Block Iterative Methods WEEK 3
  • Conservation of FD Elliptic Molecule
  • FD Elliptic Equations Summary
  • Parabolic Equations
  • 2D Parabolic Problems
  • Convergence Consistency Stability
  • Summary of Von Neumann Analysis WEEK 4
  • Hyperbolic Equations
  • Hyperbolic Time-Stepping Schemes
  • Summary of FD Solutions to PDEs
  • Intro to Finite Elements
  • FE Basis and Weighting Functions WEEK 5
  • FE Matrix Assembly
  • 2D FE Problems
  • Variable Coefficients
  • 1D Integration Formulas WEEK 6
  • Bilinear Element
  • Deformed Bilinear Element
  • Example with Bilinear Elements
  • Transient Problems on Finite Elements WEEK 7
  • FE Time-Stepping Example
  • FE Stability Analysis
  • Explicit FE Time-Stepping
  • Treatment of Point Sources
  • Summary of Finite Elements WEEK 8
  • Matrix Inversion and Inverse Noise
  • Singular Value Decomposition
  • Linear Least Squares WEEK 9
  • Fitting Models to Data
  • Constrained Minimization of Data-Model Misfit
  • Constrained Minimization Continued
  • Parameter Estimation
  • GLS Data Inversion Summary