1 The Road from Physics to Computation
This is a guided tour of the ingredients needed to turn a continuum model into a stable and verifiable numerical approximation. The focus is incompressible, viscous fluid mechanics, but many ideas apply to diffusion, elasticity, and transport problems.
1.1 Four levels to keep separate
- Physics: conservation of mass and linear momentum.
- Mathematics: differential equations and suitable function spaces.
- Discretization: mesh, finite-element spaces, and algebraic systems.
- Computation: solvers, tolerances, stability, and validation against evidence.
1.2 Why the weak form matters
The strong equations demand derivatives at every point where they are stated. A weak form transfers part of this derivative requirement to test functions and allows solutions to be represented in spaces that have fewer classical derivatives.
1.3 An important warning
In incompressible flow, pressure does more than contribute a force: it participates as a constraint variable enforcing divergence-free velocity. Discretizing velocity and pressure independently without checking compatibility can produce unstable pressure patterns.
Thought experiment: Why might a numerically small residual fail to prove that a mesh-resolved solution is physically accurate?
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.