Partial Differential Equations (PDE) and Solutions: Darcy Flow, Convection-Diffusion, Poisson Problems. In Lawrence Livermore National Laboratory (LLNL) Open Data Initiative

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.