Spectral Element Method - 2D Poisson Solver

A Julia implementation of the Spectral Element Method (SEM) for solving the 2D Poisson equation:

-∇²u = f in Ω

with Dirichlet boundary conditions.

Overview

This solver uses high-order spectral elements with Gauss-Legendre quadrature for accurate numerical solutions. The implementation features:

• Higher-order polynomial basis functions (configurable order r)
• Gauss-Legendre quadrature for numerical integration
• Structured rectangular mesh generation
• Support for Dirichlet and periodic boundary conditions
• Error computation against known analytical solutions

Files

Core Components

• Poisson.jl - Main solver implementation
• Mesh2Drect.jl - Rectangular mesh generator
• intNodes.jl - Internal node generation for higher-order elements
• ID.jl - Global degree of freedom assignment

Basis Functions

• basis.jl - Lagrange polynomial basis functions
• basisx.jl - Derivatives of basis functions
• GLpw.jl - Gauss-Lobatto points and weights computation

Matrix Assembly

• KMtrx.jl - Stiffness matrix assembly
• MMtrx.jl - Mass matrix assembly
• intWeights.jl - Integration weights computation

Utilities

• ChkNodeEx.jl - Node existence checker
• bubble_sort.jl - Sorting utility
• f_ex.jl - Source term function
• ex_sol.jl - Exact solution for error computation
• femerror.jl - Error calculation functions

Usage

1. Set problem parameters in Poisson.jl:

   # Problem size
   Nx = 4  # Number of elements in x direction
   Ny = 4  # Number of elements in y direction
   r = 4   # Polynomial order
   
   # Domain size
   Lx = 1.0  # Length in x direction
   Ly = 1.0  # Length in y direction

2. Run the solver:

   include("Poisson.jl")

3. The solution and error metrics will be computed and displayed.

Known issues

• The degree of freedom (DOF) generation process in ID.jl and ChkNodeEx.jl currently uses a naive approach that:

  1. Assigns DOFs sequentially to all nodes
  2. Uses ChkNodeEx.jl to check for existing nodes through direct array comparisons
  3. Then searches for and eliminates duplicate DOFs using nested loops This becomes a significant bottleneck for large meshes or high polynomial orders.

• The DOF assignment and node checking could be optimized by:

  • Using more efficient data structures (e.g., hash sets) to track unique DOFs and nodes
  • Implementing a smarter initial assignment that avoids duplicates
  • Replacing the direct array comparisons in ChkNodeEx.jl with spatial partitioning
  • Parallelizing the duplicate detection process

This node/DOF handling is currently the main performance bottleneck when scaling to larger problems or higher orders.

License

This code is released under the MIT License. See LICENSE file for details.