Mathbridge

FEM is the cornerstone of modern CAE. Did you know FEM can also tackle CFD? Absolutely! As industrial challenges become interdisciplinary, FEM-based CFD is rising in relevance due to its ability of easily handling Multiphysics. 

Think of CAE tools as languages—FEM/variational method is the universal *script*. Rooted in mathematics, it bridges disciplines and methods seamlessly. 

Whether you’re in geology, medicine, design, chemistry, architecture, or STEM at large: FEM is for you (and so is this course!). While FEM’s math-heavy reputation intimidates many, **this course cuts through complexity**. No advanced math required—just undergrad calculus, differential equations, and linear algebra. 

**In two hours, you’ll gain intuitive, hands-on skills to.

In just two hours, you will:

• Build spreadsheet-powered FEM solvers for practically relevant problems like heat transfer/diffusion with reaction and basic flow problems. No coding required—everything is laid out step-by-step in Excel so you see exactly how each matrix entry and load vector arises.
• Tame convection-dominated flows with SUPG stabilization. One of the challenges in FEM CFD has been modelling  high-speed flow problems accurately. Implement the Streamline-Upwind/Petrov–Galerkin approach, and watch the change in matrix structures in real time as it removes unphysical oscillations from your solutions.
• Understand both conservation and variational frameworks. Conservation equations and balance equations put the differential equations into correct perspective! Derive finite-volume flux balances and finite-element weak forms side-by-side—so you’ll recognize the mathematical underpinnings, diffusion, and transport in any CFD package. Avoid ill posed unsolvable equations by constructing these balances from scratch.
• Understand basic error analysis & convergence plotting. Calculate L₂ error norms, generate log–log convergence plots, and develop the critical habit of verifying your mesh refinement improves accuracy.

Extra Content: Access a Julia script that reads Gmsh meshes, solves 1D Poisson and SUPG problems, and runs a grid-convergence study via the Method of Manufactured Solutions across various polynomial orders. This optional module acts as a bridge towards full unstructured 2D Navier Stokes coding.

By course end, you’ll not only be able to build a solid foundation towards tackling practical industrially relevant problems using FEM, but also begin to understand what goes on behind the scenes of commercial solvers—empowering you to get better at using black boxes, better at troubleshooting results, as you confidently tackle more advanced multiphysics simulations.

Who this course is for:

Engineers who want a quick dive in FEM basics from a transport phenomena angle

Undergraduate and graduate students from STEM fields who have completed their undergraduate math courses.

Researchers & industry professionals who use FEM/FEA softwares and want to understand the basics (from STEM background needed)