Discrete Exterior Calculus (DEC) is a mathematical framework used to discretize and solve differential equations on discrete spaces, such as meshes. It combines concepts from differential geometry, algebraic topology, and numerical analysis to provide a unified approach to solving problems in physics, engineering, and computer graphics.
FEEC stands for Finite Element Exterior Calculus, which is a variant of DEC that is specifically designed for solving problems on meshes with triangular or tetrahedral elements. It uses a finite element method to approximate the solution to a differential equation on a discrete space.
DEC and FEEC have a wide range of applications in various fields such as computational fluid dynamics, computer graphics, and geometric modeling. They are also used in physics and engineering to solve problems involving partial differential equations, such as heat transfer, fluid flow, and elasticity.
There are several books, research papers, and online resources available for learning about DEC and FEEC. Some recommended books include "Discrete Exterior Calculus" by Hirani and "Finite Element Exterior Calculus" by Arnold, Falk, and Winther. Online resources include lecture notes, tutorials, and video lectures on websites such as YouTube and arXiv.
DEC and FEEC offer several advantages over traditional methods for solving differential equations. They provide a unified framework for solving problems on discrete spaces, which makes them applicable to a wide range of problems. They also offer better accuracy and efficiency, especially for problems defined on curved or irregular domains.