Iterative Methods for Elliptic : Laplace and Poisson Solvers

The chapter provides methods for selecting solvers, analyzing performance, and applying these techniques to pressure and scalar field computations in CFD. This chapter examines iterative methods for solving elliptic partial differential equations, focusing on the Laplace and Poisson equations in two dimensions. It covers mathematical formulation, boundary condition treatment, and discretization using finite difference methods. Iterative solvers including Jacobi, Gauss–Seidel, and Successive Over-Relaxation (SOR) are applied to the discrete systems, and their convergence behavior is evaluated using L 2 error norms. The chapter outlines matrix storage formats such as CSR and CSC and discusses solver choices involving direct methods, Krylov subspace techniques, and multigrid algorithms. Preconditioning strategies are introduced to improve convergence. Pressure referencing approaches are discussed in the context of incompressible flow to ensure unique solutions. Multigrid cycles are presented to illustrate solver acceleration across grid levels.