Chapter 5: Numerical Solution of Algebraic Systems

Overview

This chapter covers methods for solving systems of linear and nonlinear algebraic equations arising from discretized PDEs.

Learning Objectives

  • Implement direct solution methods
  • Apply iterative solution techniques
  • Understand preconditioning strategies
  • Solve nonlinear systems

Topics

5.1 Direct Methods (Gaussian Elimination, LU Decomposition)

5.2 Iterative Methods (Jacobi, Gauss-Seidel, SOR)

5.3 Krylov Subspace Methods

5.4 Preconditioning Techniques

5.5 Nonlinear Systems (Newton-Raphson)