10  Toward Full Incompressible-Flow Solvers

The planned progression is incremental. The goal is to make every additional ingredient accountable to a mathematical or physical requirement.

10.1 The next build stages

1. Scalar finite elements: stiffness, loads, essential boundary conditions.
2. Vector-valued spaces and Stokes block assembly.
3. Pressure normalization and stable mixed element pairs.
4. Solver selection and block preconditioning.
5. Convection, nonlinear residuals, and time integration.
6. Stabilization at higher Reynolds numbers, with explicit assumptions.
7. Adaptive mesh refinement and a posteriori error indicators.
8. Verification benchmarks with convergence tables.

The R appendix starts at stage one. A later edition can use specialized finite-element libraries for the mixed multidimensional systems while retaining R for experiments, visualization, and data analysis.

Note

Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.