6 Finite-Element Spaces and Assembly
6.1 From a mesh to algebra
A mesh partitions the domain into elements. Velocity and pressure fields are represented in finite-dimensional trial spaces by element basis functions and unknown coefficients.
On each element \(\Omega_e\), evaluate integrals that contribute to the global matrices. Assembly combines local contributions using the mapping from local degrees of freedom to global indices.
6.2 Quadrature and geometry
Finite-element integration requires a relationship between a reference element and the physical element. The Jacobian controls the transformation of derivatives and integration measures. Inadequate quadrature, poor element shapes, and inconsistent geometry can compromise accuracy.
6.3 The mixed nature of incompressibility
A stable velocity-pressure pair is not simply one with many degrees of freedom. The choice of polynomial orders and function spaces determines how well the constraint is represented.
A common example is the Taylor–Hood pairing of continuous quadratic velocity with continuous linear pressure on suitable meshes. Equal-order pairs often require special stabilization, and convergence claims depend on the formulation and problem setting.
Numerical exercise: Assemble the stiffness matrix for a one-dimensional linear-element diffusion problem before attempting the mixed Stokes blocks.
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.