Laplacian in 1D, 2D, or 3D

Sparse (1-3)D Laplacian on a rectangular grid with exact analytic eigenpairs.

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Laplacian Eigenpair Computation

This MATLAB/Octave-compatible code computes analytically exact eigenpairs of the negative Laplacian operator in 1D, 2D, or 3D on a rectangular finite-difference grid. It supports a wide range of boundary condition combinations: Dirichlet (D), Neumann (N), and Periodic (P).

The first mandatory output is the sparse Laplacian matrix itself via Kronecker sums of 1D discrete Laplacians. The actual numerical entries of the matrix fit int8 format, but only double data class is yet supported for sparse matrices in MATLAB.

References

• Eigenvalues and eigenvectors of the second derivative
• Kronecker sum of discrete Laplacians

Examples

Compute both matrix and eigenpairs for a 3D Laplacian with mixed boundary conditions:

[A, lambda, V] = laplacian([100, 45, 55], {'DD', 'NN', 'P'}, 20);

Compute only the matrix:

A = laplacian([100, 45, 55], {'DD', 'NN', 'P'});

Features

• Supports Dirichlet, Neumann, and Periodic BCs (including mixed combinations)
• Computes eigenvalues and eigenvectors analytically using Kronecker structure
• Builds the Laplacian matrix directly
• Compatible with GNU Octave

Related Projects. This code is a part of the BLOPEX eigensolver package:

• Wikipedia: BLOPEX
• GitHub BLOPEX Repository

Notes

The Python multidimensional code in SciPy:

• LaplacianNd — SciPy v1.16.0 Manual

supports arbitrary dimensions, but it does not yet support mixed boundary conditions.

Authors

• Bryan C. Smith
• Andrew V. Knyazev