Finite Element Method Source Code for One-Dimensional Singular Perturbation Problems (Convection-Diffusion Problems)

The following content presents a finite element method source code for one-dimensional singular perturbation problems (convection-diffusion problems), which can simulate heat transfer, mass transfer, or momentum transfer processes. The code implementation includes multiple functions and subroutines for calculating various physical quantities such as flux, temperature gradients, and concentration gradients. The program incorporates several numerical methods including Euler's method, implicit schemes, and the Crank-Nicolson method, with configurable options for time step sizes and mesh resolution to determine computational intervals. This program employs the finite element method - a numerical technique for solving differential equations, particularly partial differential equations. The method discretizes the domain into multiple small subregions, each associated with a nodal point. At each node, local basis functions are used to approximate the solution. This approach transforms differential equations into systems of algebraic equations that can be solved computationally through matrix operations and linear algebra routines. The source code implements key algorithms including: / Galerkin formulation for weak form derivation / Element-wise matrix assembly for stiffness and mass matrices / Upwind stabilization techniques for handling convection-dominated cases / Adaptive time-stepping algorithms for numerical stability Thus, this finite element source code for one-dimensional singular perturbation problems serves as a valuable tool for scientists and engineers to better understand and simulate various physical processes, with particular emphasis on robust numerical implementation and computational efficiency.