NumPDE

Overview

NumPDE is a comprehensive collection of numerical algorithms developed for the course Numerical Methods for Partial Differential Equations at ETH Zurich’s Computational Science and Engineering (CSE) department. The repository serves as both a teaching tool and a reference implementation for modern PDE discretization techniques.

Key Features

Broad Coverage of Physical Phenomena

The repository includes ~100 algorithms spanning a wide range of applications:

• Elliptic problems: Elastic membranes, electrostatics, diffusion
• Evolution problems: Heat equation, wave propagation
• Convection-diffusion: Upwinding, streamline diffusion, semi-Lagrangian methods
• Conservation laws: Traffic flow, shock waves, Riemann problems
• Fluid dynamics: Stokes equations, saddle-point problems
• Electromagnetism: Maxwell’s equations, Whitney forms

Numerical Methods

Implementations cover the core techniques taught in graduate-level numerical analysis:

• Finite Element Methods (FEM): Lagrangian and parametric elements, Galerkin discretization
• Finite Volume Methods (FVM): Conservative schemes, numerical fluxes, MUSCL
• Finite Difference Methods (FDM): Classical discretizations
• Timestepping: Runge-Kutta methods, implicit schemes for stiff problems
• Error analysis: A priori estimates, convergence verification

Educational Design

Each algorithm is structured for clarity and pedagogical value, with emphasis on:

• Clean, readable C++ implementations using LehrFEM++
• Modular design for easy modification and experimentation
• Accompanying lecture notes and practice problems

Resources

• Lecture Notes
• Practice Problems

Background

This project was co-developed during my time as a PhD student and scientific assistant at ETH Zurich (2018-2022), in collaboration with Prof. Ralf Hiptmair and fellow teaching assistants. It remains the standard framework for the Numerical Methods for PDEs course, which I now teach as a lecturer starting in 2026.