Is this project an undergraduate, graduate, or faculty project?
Undergraduate
Project Type
individual
Campus
Daytona Beach
Authors' Class Standing
Logan Price, Senior
Lead Presenter's Name
Logan Price
Lead Presenter's College
DB College of Arts and Sciences
Faculty Mentor Name
Hemanta Kunwar
Abstract
Numerical Methods for Nonlinear Problems Using the Finite Element Method is a computational mathematics capstone that builds and tests finite element method (FEM) workflows for nonlinear partial differential equations in FreeFEM++, with ParaView used for visualization. Two nonlinear model problems are used to demonstrate the approach. The first is a semilinear reaction-diffusion equation with a cubic nonlinearity. A manufactured solution is used so accuracy can be checked at a fixed final time, and refinement studies in both time step and mesh size are run while nonlinear iteration counts are tracked to show solver effort. The second problem is the steady incompressible Navier-Stokes equations for flow past a circular obstacle. A Newton-type method is used, and a continuation strategy in viscosity (Reynolds number) is applied to improve robustness as the flow becomes more nonlinear. Together, these problems highlight the full FEM pipeline: weak forms are set up, nonlinear systems are assembled and solved, convergence behavior is monitored, and VTU outputs are exported for consistent ParaView figures. The main outcome is a reproducible set of scripts, plots, and visualizations that show how nonlinear PDEs can be solved and analyzed using FEM in FreeFEM++.
Did this research project receive funding support (Spark, SURF, Research Abroad, Student Internal Grants, Collaborative, Climbing, or Ignite Grants) from the Office of Undergraduate Research?
No
Included in
Non-linear Dynamics Commons, Numerical Analysis and Computation Commons, Special Functions Commons
Numerical Methods for Nonlinear Problems Using the Finite Element Method
Numerical Methods for Nonlinear Problems Using the Finite Element Method is a computational mathematics capstone that builds and tests finite element method (FEM) workflows for nonlinear partial differential equations in FreeFEM++, with ParaView used for visualization. Two nonlinear model problems are used to demonstrate the approach. The first is a semilinear reaction-diffusion equation with a cubic nonlinearity. A manufactured solution is used so accuracy can be checked at a fixed final time, and refinement studies in both time step and mesh size are run while nonlinear iteration counts are tracked to show solver effort. The second problem is the steady incompressible Navier-Stokes equations for flow past a circular obstacle. A Newton-type method is used, and a continuation strategy in viscosity (Reynolds number) is applied to improve robustness as the flow becomes more nonlinear. Together, these problems highlight the full FEM pipeline: weak forms are set up, nonlinear systems are assembled and solved, convergence behavior is monitored, and VTU outputs are exported for consistent ParaView figures. The main outcome is a reproducible set of scripts, plots, and visualizations that show how nonlinear PDEs can be solved and analyzed using FEM in FreeFEM++.