math-fem

Multigrid FEM solver for the Helmholtz equation.

Installation

Add to your Cargo.toml:

[dependencies]
math-fem = { path = "../math-fem" }

Features

• 2D and 3D meshes: Triangles, quadrilaterals, tetrahedra, hexahedra
• Lagrange elements: P1, P2, P3 polynomial basis functions
• Boundary conditions: Dirichlet, Neumann, Robin, PML
• Multigrid solver: V-cycle, W-cycle with geometric coarsening
• Adaptive refinement: h-refinement with residual-based error estimation

Usage

use fem::{mesh, boundary::BoundaryConditions};

// Create a 2D mesh
let mesh = mesh::unit_square_triangles(10);

// Create a 3D mesh
let mesh_3d = mesh::unit_cube_tetrahedra(5);

Modules

mesh

Mesh generators for common domains:

use fem::mesh::*;

// 2D meshes
let rect = rectangular_mesh_triangles(0.0, 1.0, 0.0, 1.0, 10, 10);
let quads = rectangular_mesh_quads(0.0, 1.0, 0.0, 1.0, 10, 10);
let circle = circular_mesh_triangles(0.0, 0.0, 1.0, 5, 16);
let annulus = annular_mesh_triangles(0.0, 0.0, 0.5, 2.0, 4, 16);

// 3D meshes
let box_tet = box_mesh_tetrahedra(0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 5, 5, 5);
let box_hex = box_mesh_hexahedra(0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 5, 5, 5);

// Convenience functions
let square = unit_square_triangles(10);
let cube = unit_cube_tetrahedra(5);

basis

Lagrange polynomial basis functions:

• P1, P2, P3 for simplices (triangles, tetrahedra)
• Q1, Q2 for quads/hexes

boundary

Boundary condition handling:

use fem::boundary::BoundaryConditions;
use num_complex::Complex64;

let mut bcs = BoundaryConditions::new();

// Dirichlet: u = g on boundary
bcs.add_dirichlet(1, |x, y, z| Complex64::new(1.0, 0.0));

// Neumann: du/dn = h on boundary
bcs.add_neumann(2, |x, y, z| Complex64::new(0.0, 0.0));

// Robin: du/dn + alpha*u = g on boundary
bcs.add_robin(3,
    |x, y, z| Complex64::new(1.0, 0.0),  // alpha
    |x, y, z| Complex64::new(0.0, 0.0),  // g
);

assembly

Matrix assembly for FEM:

• Stiffness matrix assembly
• Mass matrix assembly
• Helmholtz matrix (K - k²M)

quadrature

Gaussian quadrature rules for numerical integration.

multigrid

Geometric multigrid solver:

• V-cycle, W-cycle, F-cycle methods
• Gauss-Seidel smoothing
• Linear interpolation transfer operators

Element Types

Feature Flags

• native (default) - Enables rayon parallelism and native BLAS
• parallel - Enables rayon for parallel assembly

Dependencies

• math-solvers - Sparse linear algebra solvers
• ndarray - N-dimensional arrays
• num-complex - Complex number support
• rayon (optional) - Parallel processing