Ponce, Colin; Li, Ruipeng; Mao, Christina; Fairbanks, Hilary (2022). Partial Differential Equations (PDE) and Solutions: Darcy Flow, Convection-Diffusion, Poisson Problems. In Lawrence Livermore National Laboratory (LLNL) Open Data Initiative. UC San Diego Library Digital Collections. https://doi.org/10.6075/J0HM58MK

These data represent inputs and solutions to Darcy flow, convection-diffusion, and Poisson problems over 2D square domains. Scripts to generate the datasets are included.

Convection-Diffusion and Poisson Dataset
These two datasets contain 10,000 steady-state convection-diffusion and pure diffusion problems, respectively, on the 2-D unit square.
The problems were discretized by finite element methods (FEM) using [1,2,4]. The input data include the (convection-) diffusion coefficients, mesh point coordinates and heat source (the right-hand sides of the equations). The target data are the numerical solutions obtained by using [3] and the flux of the heat.

* [1] FEniCS open source PDE solver https://fenicsproject.org/
* [2] DOLFIN C++/Python interface of FEniCS https://pypi.org/project/DOLFIN/
* [3] Numpy open-source Python library https://numpy.org/
* [4] Sympy open-source Python library https://www.sympy.org/en/index.html

Darcy Flow Data
This dataset contains finite element solutions of the Darcy Flow equation.
Numerical simulations were performed using tools developed in ParELAG[4], a parallel C++ library for performing numerical upscaling of finite element discretizations and AMG techniques, and ParELAGMC [5], a parallel element agglomeration MLMC library. These libraries use MFEM [3] to generate the fine grid finite element discretization and HYPRE [2] to handle massively parallel linear algebra.

* [1] GLVis: Opengl finite element visualization tool. glvis.org.
* [2] HYPRE: High performance preconditioners. http://www.llnl.gov/CASC/hypre/.
* [3] MFEM: Modular finite element methods library. mfem.org.
* [4] ParELAG: Element-agglomeration algebraic multigrid and upscaling library, version 2.0. http://github.com/LLNL/parelag, 2015.
* [5] ParELAGMC: Parallel element agglomeration multilevel Monte Carlo library. http://github.com/LLNL/parelagmc, 2018.

• 2021

• 2022

• Ponce, Colin

• Li, Ruipeng

• Fairbanks, Hilary

• Mao, Christina

The datasets are in .npz format and can be accessed in your Python program with `numpy.load` module or with a desktop viewer [npzviewer](https://pypi.org/project/npzviewer/).

This work was produced under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and was supported by the LLNL-LDRD program under Project No. 19-ERD-019.

• Convection-diffusion problem
• Darcy Flow problem
• Finite element method
• Machine learning
• Multilevel Neural Network (MTNN)
• Numerical problem
• Partial differential equation
• Poission problem

• data

• No linguistic content; Not applicable

Christina Mao: https://orcid.org/0000-0001-9435-4261

Colin Ponce: https://orcid.org/0000-0002-6720-8805

Ruipeng Li: https://orcid.org/0000-0003-2802-5763

Doi: https://doi.org/10.6075/J0HM58MK

Software

• DOLFIN C++/Python interface of FEniCS: https://pypi.org/project/DOLFIN/
• FEniCS open source PDE solver: https://fenicsproject.org/
• GLVis: Opengl finite element visualization tool: https://glvis.org/
• HYPRE: High performance preconditioners: https://www.llnl.gov/CASC/hypre/
• MFEM: Modular finite element methods library: https://mfem.org/
• Numpy open-source Python library: https://numpy.org/
• ParELAG: Element-agglomeration algebraic multigrid and upscaling library, version 2.0. 2015: https://github.com/LLNL/parelag
• ParELAGMC: Parallel element agglomeration multilevel Monte Carlo library. 2018: https://github.com/LLNL/parelagmc
• Sympy open-source Python library: https://www.sympy.org/

Reference

• Multilevel Neural Network (MTNN) Library on GitHub: https://github.com/LLNL/MTNN

Creative Commons Attribution 4.0 International Public License

• Lawrence Livermore National Laboratory

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2022-07-13