Laplacian in 1D, 2D, or 3D

版本 1.3 (28.8 KB) 作者: Andrew Knyazev
Sparse (1-3)D Laplacian on a rectangular grid with exact analytic eigenpairs.
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更新时间 2025/6/24

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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
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:
Notes
The Python multidimensional code in SciPy:
supports arbitrary dimensions, but it does not yet support mixed boundary conditions.
Authors
  • Bryan C. Smith
  • Andrew V. Knyazev

引用格式

Andrew Knyazev (2025). Laplacian in 1D, 2D, or 3D (https://ww2.mathworks.cn/matlabcentral/fileexchange/27279-laplacian-in-1d-2d-or-3d), MATLAB Central File Exchange. 检索时间: .

MATLAB 版本兼容性
创建方式 R2010b
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版本 已发布 发行说明
1.3

Corrected version number.
Cosmetic changes in the code.
Updated the description to reflect the change of the order of the output made in the previous revision and other updates.
1.2

Revision 1.1 changes: rearranged the output variables, always compute the eigenvalues, compute eigenvectors and/or the matrix on demand only.
added a conversion to a toolbox
1.1.0.0

updated description
1.0.0.0