Finite Element Solvers for Advection-Diffusion and Stokes Problems

This repository contains MATLAB implementations for the numerical solution of partial differential equations (PDEs) related to fluid dynamics. The project is divided into two main parts: a 2D Convection-Diffusion solver with upwind stabilization and a Stokes flow solver using different stable/stabilized FEM elements.

Features

1. Convection-Diffusion Solver (Project 1)

Solves the steady-state Advection-Diffusion equation on a unit square domain .

• Method: Galerkin Finite Element Method (FEM) with linear triangular elements.
• Stabilization: Streamline Upwind Diffusion (SUD) to handle high Peclet numbers and prevent numerical oscillations.
• Boundary Conditions: Dirichlet conditions handled via boundary lifting operators.
• Scenarios: Tested across different regimes, from pure diffusion to convection-dominated flows ().

2. Stokes Equation Solver (Project 2)

Solves the incompressible Stokes equations for velocity and pressure .

• Algorithms Implemented:

1. P1/P1: Standard linear elements for both velocity and pressure (unstable, used for comparison).
2. P1/P1 GLS (Galerkin Least Squares): Stabilized formulation using a pressure-stabilizing term ().
3. Mini-Element (Bubble Scheme): Enrichment of the velocity space with cubic bubble functions to satisfy the Info-Sup (LBB) condition.

• Test Case: Validated on the Lid-Driven Cavity problem.
• Error Analysis: Comparison of residual errors across multiple mesh refinements.

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Implementation Details

The code is modularized into specialized functions to ensure clarity and reusability:

Function	Description
input_var.m	Parses mesh data and connectivity from the /mesh folder.
LocalB.m	Evaluates basis functions, gradients, and element areas.
MatrixB.m	Builds global Stiffness (), Transport (), and Stabilization () matrices.
lifting.m	Applies the boundary lifting operator for non-homogeneous Dirichlet BCs.
stokes_solver.m	Unified function for P1/P1, GLS, and Bubble schemes.

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Key Results

Convection-Diffusion Stability

When the diffusion coefficient is small, the standard Galerkin method produces spurious oscillations. The SUD stabilization effectively reduces these artifacts, allowing for accurate solutions even at high Peclet numbers.

Stokes Convergence

The P1/P1 GLS and Bubble schemes provide stable pressure fields, unlike the standard P1/P1 approach which suffers from pressure instability.

• GLS Tuning: Observations show that is optimal for the Lid-Driven Cavity; higher values lead to excessive smoothing of the solution.

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How to Run

1. Clone the repository.
2. Add the mesh folders to your MATLAB path.
3. Execute the main scripts:

• fem_solver.m for Convection-Diffusion analysis.
• Project2_CS.m for Stokes flow simulations.
• Project2_CS_LidCavityPb.m for Lid Cavity problem.

4. The results will automatically display plots for velocity components (), pressure (), and error convergence.