Stokes Equations

$$ \begin{cases} -\nu \Delta u + \nabla p = 0 & \text{in } \Omega \\ \nabla \cdot u = 0 & \text{in } \Omega \ u = g & \text{on } \Gamma_D \\ -pn + \nu (\nabla u)n = h & \text{on } \Gamma_N \end{cases} $$

$$ \begin{cases} \int_{\Omega} \nu \nabla u : \nabla v , d\Omega - \int_{\Omega} p (\nabla \cdot v) , d\Omega = \int_{\Gamma_N} h \cdot v , d\Gamma - \int_{\Omega} \nu \nabla r_g : \nabla v , d\Omega & \forall v \in X \\ -\int_{\Omega} q (\nabla \cdot u) , d\Omega = \int_{\Omega} q (\nabla \cdot r_g) , d\Omega & \forall q \in Q \end{cases} $$

This project implements a finite element discretization of the stationary incompressible Stokes equations in two spatial dimensions. The formulation is based on the mixed velocity-pressure weak form with Dirichlet and Neumann boundary conditions.

The implementation includes:

• Assembly of the velocity stiffness matrix $A$
• Construction of the pressure coupling matrices $B_x$ and $B_y$
• Formation of the saddle-point system
• Handling of Dirichlet lifting functions
• Structured sparse matrix assembly using scipy.sparse
• Mesh refinement and orientation correction utilities
• Visualization tools for:
  • sparse matrix structures,
  • streamline fields,
  • refined finite element meshes

The numerical discretization uses triangular finite elements and constructs the global system through element-wise assembly of local contributions derived from affine reference mappings.

The saddle-point system has the block structure

$$ K = \begin{bmatrix} A & 0 & B_x^T \\ 0 & A & B_y^T \\ B_x & B_y & 0 \end{bmatrix} $$

where $A$ denotes the discrete vector Laplacian and $B_x, B_y$ represent the discrete divergence operators.

The repository additionally contains routines for post-processing and visualization of the computed velocity field, including streamline interpolation on unstructured meshes.