MPS4PDEs

Minimal Positive Stencils for Partial Differential Equations

This repository provides implementations of the Minimal Positive Stencil (MPS) method for meshfree discretization of linear elliptic partial differential equations in non-divergence form via nonlocal relaxation, developed in the paper:

Ye, Q., Tian, X. Monotone Meshfree Methods for Linear Elliptic Equations in Non-divergence Form via Nonlocal Relaxation. J Sci Comput 96, 85 (2023).
https://doi.org/10.1007/s10915-023-02294-3

The MPS framework constructs monotone meshfree discretizations by enforcing nonnegativity and consistency constraints, enabling provably sparse and convergent schemes for non-divergence form PDEs without requiring a mesh structure. The approach is particularly well-suited for high-dimensional or irregular domains where mesh-based methods are difficult to apply.

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Repository Structure

The repository contains two main subdirectories:

• 2d_test — MATLAB implementation for two-dimensional test problems.
• 3d_test — MATLAB implementation for three-dimensional test problems.

Each subdirectory includes scripts and MATLAB classes for defining domains, equations, and coefficient matrices, along with example workflows for testing manufactured solutions and generating error analyses.

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Requirements

• MATLAB R2019b or later (argument validation features are required).
• The code is not fully optimized; some data structures are more efficiently implemented in compiled languages such as C++.

If you are interested in contributing or rewriting the code in other languages (e.g., C++), please contact Qihao Ye (q8ye@ucsd.edu)

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Quick Start

Each dimension-specific folder provides a step-by-step pipeline for running test problems.

Example Workflow (2D and 3D)

1. Initialize point cloud: Run test_step01_initialize_point_cloud with a specified domain.
2. Assign stiffness matrices: Run test_step02_b_assign_checkpoint, requiring outputs from Step 01, the domain, and matrixA class.
3. Solve equation and compute errors: Run test_step03_solve_equation, requiring Step 02 outputs, the domain, matrixA, and equation.
4. Plot error graphs: Run test_step04_error_graph using Step 03 outputs to visualize numerical errors.

Each class component is modular:

• C_Domain: domain geometry and boundary handling.
• C_Equation: manufactured solutions and differential operators.
• C_MatrixA: coefficient matrices defining PDE parameters.
• C_PointCloud: handles the generation, organization, and management of point cloud data.

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Citation

If you find this repo useful or use this repo in your research, please cite:

@article{ye2023monotone,
    title={Monotone Meshfree Methods for Linear Elliptic Equations in Non-divergence Form via Nonlocal Relaxation},
    author={Ye, Qihao and Tian, Xiaochuan},
    journal={Journal of Scientific Computing},
    volume={96},
    number={3},
    pages={85},
    year={2023},
    publisher={Springer}
}

Resources:
[Project Website] • [Journal Version] • [Correction] • [arXiv Version]