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An experimental set of finite element flow solvers for heterogeneous (GPU/CPU) systems. The initial development of libParanumal was performed by the Parallel Numerical Algorithms Group at Virginia Tech.

libParanumal is funded in part by the US Department of Energy as part of the activities of the Center for Efficient Exscale Discretizations.

Why is it called libParanumal?: the high-order finite-element implementations in libParanumal are spectrally accurate and rely heavily on ghost elements for MPI communications.

1. How to cite the libParanumal project:

If you use any part of libParanumal in your research project including variants on the included compute kernels please use the following bibliographical reference:

      author = "Chalmers, N. and Karakus, A. and Austin, A. P. and Swirydowicz, K. and Warburton, T.",
      title = "{libParanumal}: a performance portable high-order finite element library",
      year = "2020",
      url = "",
      doi = "10.5281/zenodo.4004744",
      note = "Release 0.4.0"

see the references section below for additional papers to reference about various aspects of the library.

2. How you can help out:

libParanumal is a community project. Please help improve the library by submitted an Issue if you notice any unexpected behavior, discover a bug, have problems installing/running/trouble shooting your installation. It benefits us as a community when issues and feature requests are shared so that we can understand how to improve the library.

Please submit feature requests as an Issue for the consideration of all contributors. Likewise if you wish to submit a code change please make a GitHub Pull Request.

3. Overview

Brief summary of major features:

A. Supported elements:

  • Triangles, quadrilaterals, tetrahedra, hexahedra.
  • Lagrange basis functions.

B. Mesh wrangling:

  • Gmsh format file loaders.
  • Load balanced geometric partitioning using space filling curves (Hilbert or Morton ordering).

C. Time integrators:

  • Adaptive rate Dormand-Prince order 5 Runge-Kutta.
  • Low storage explicit Runge-Kutta order 4.
  • Single and Multirate Adams-Bashforth order 3.
  • Extrapolated Backwards Differencing order 3.

D. Iterative linear solvers:

  • Preconditioned (flexible) Conjugate Gradient method.
  • Non-blocking Preconditioned (flexible) Conjugate Gradient method.
  • Preconditioned (Generalized) Minimal Residual Method.
  • Improved initial guess techniques (History Projection/Extrapolation).

E. Elliptic solver:

  • Linear Poisson and screened Poisson potential solvers.
  • GPU-optimized matrix-vector products.
  • p-type multigrid, algebraic multigrid, low-order SEMFEM, Overlapping Additive Schwarz, and Jacobi preconditioning.
  • Matrix-free p-multigrid for fine levels of multigrid hierarchy.

F. Heterogeneous accelerated flow solvers:

  • Compressible Navier-Stokes solver with:
    • Upwind discontinuous Galerkin discretization in space.
    • Optional isothermal equation of state.
  • Galerkin-Boltzmann gas dynamics solver with:
    • Penalty flux DG discretization in space.
    • Adaptive semi-analytic (pointwise exponential) integration in time.
    • Multiaxial quasi-perfectly matched absorbing layer far field boundary condition.
  • Incompressible Navier-Stokes solver with:
    • Choice of continuous FEM or interior penalty DG in space.
    • Extrapolation-BDF integration in time.
    • Sub-cycling (Operator Integration Factor Splitting) for advection.

G. Dependencies:

  • Message Passing Interface (MPI v3.0 or higher).
    • The libParanumal makefiles assume that mpic++ is installed and visible in your path.
  • Open Concurrent Compute Abstraction (OCCA)
    • OCCA must be installed.
    • OCCA will try to detect if any of these execution models are installed: OpenMP, CUDA, OpenCL, and/or HIP.
    • By default, if OCCA does not detect a chosen mode of execution it will default to Serial execution.
    • You will need to adjust the libParnumal setup input files to choose the execution model and compute device appropriate for your system.
    • The OCCA github repo is here
    • The OCCA webpage is here

4. Code block diagram

5. OCCA dependency

OCCA is held as a git submodule inside libParanumal. If you did not clone with --recursive then run the following command before building. git submodule init git submodule update

6. Required Libraries

libParanumal requires installed BLAS and LAPACK libraries. By default, the build system will look for libblas and liblapack in your default library search paths. The library paths can also be manually specified in with the LIBP_BLAS_DIR and LIBP_LAPACK_DIR variables.

Some Linux distributions will package BLAS and LAPACK libraries. For example, on Ubuntu systems these libraries can be installed via sudo apt install libblas-dev liblapack-dev

libParanumal also depends on the gslib library for gather-scatter operations. For more information on gslib see Henry Tufo's thesis and a more recent reference Fischer et al.. The source code for gslib is included in this repository.

7. Clone: libParanumal

git clone

7-1. Build all libParanumal solvers

cd libparanumal make -j `nproc`

8. Running the codes:

Each solver resides in its respective sub-directory in solvers/. Each solver sub-directory includes makefile, src directory, data directory (including header files for defining boundary conditions), okl kernel directory, and setups directory. The setups directory includes a number of example input files that specify input parameters for the solver.

8-1. Build libParanumal elliptic solver

cd libparanumal/solvers/elliptic make -j `nproc`

8-2. Run elliptic example with provided quadrilateral set up file on a single device:

./ellipticMain setups/setupQuad2D.rc

8-3. Run the same example with four devices:

mpiexec -n 4 ./ellipticMain setups/setupQuad2D.rc

9. License

The MIT License (MIT)

Copyright (c) 2017-2021 Tim Warburton, Noel Chalmers, Jesse Chan, Ali Karakus

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.


10. References

Entroy Stable DG schemes for compressible Navier-Stokes (cns): arXiv version: Chan, J., Lin, Y. and Warburton, T., 2020. Entropy stable modal discontinuous Galerkin schemes and wall boundary conditions for the compressible Navier-Stokes equations.

Initial Guesses for Sequences of Linear Systems: arXiv version: Austin, A.P., Chalmers, N. and Warburton, T., 2020. Initial Guesses for Sequences of Linear Systems in a GPU-Accelerated Incompressible Flow Solver.

Portable Streaming kernels: arXiv version: Chalmers, N. and Warburton, T., 2020. Portable high-order finite element kernels I: Streaming Operations.

Massive parallel nodal DG simulator for (acoustics): preprint: Melander, A., Strøm, E., Pind, F., Engsig-Karup, A., Jeong, C.H., Warburton, T., Chalmers, N., and Hesthaven, J.S., 2020. Massive parallel nodal discontinuous Galerkin finite element method simulator for room acoustics.

Scalability of high-performance PDE solvers: publisher, arXiv version: Fischer, P., Min, M., Rathnayake, T., Dutta, S., Kolev, T., Dobrev, V., Camier, J.S., Kronbichler, M., Warburton, T., Swirydowicz, K. and Brown, J., 2020. Scalability of high-performance PDE solvers, The International Journal of High Performance Computing Applications, Vol 34, Issue 5, Pages 562-586.

Discontinuous Galerkin Boltzmann (bns) solver: publisher, arXiv version: Karakus, A., Chalmers, N., Hesthaven, J.S. and Warburton, T., 2019. Discontinuous Galerkin Discretizations of the Boltzmann Equations in 2D: semi-analytic time stepping and absorbing boundary layers, Journal of Computational Physics, Volume 390, Pages 175–202.

Incompressible Navier-Stokes (discontinuous) Galerkin (ins) solver: publisher, arXiv version: Karakus, A., Chalmers, N., Swirydowicz, K. and Warburton, T., 2019. A GPU accelerated discontinuous Galerkin incompressible flow solver, Journal of Computational Physics, Volume 390, Pages 380–404, 2019.

Optimization of elliptic mat-vec operations for (elliptic) solver on hexes: publisher, arXiv version: Swirydowicz, K., Chalmers, N., Karakus, A., and Warburton, T. 2019. Acceleration of tensor-product operations for high-order finite element methods, The International Journal of High Performance Computing Applications, Vol 33, Issue 4.

Low-order preconditioning of triangular elements (elliptic precon): publisher: Chalmers, N. and Warburton, T. 2018. Low-order preconditioning of high-order triangular finite elements, SIAM Journal on Scientific Computing, Vol 40, Issue 6, Pages A4040-A4059

11. Technical Reports

CEED MS36: link: Kolev, T., Fischer, P., Austin, A.P., Barker, A.T., Beams, N., Brown, J., Camier, J.-S., Chalmers, N., Dobrev, V., Dudouit, Y., Ghaffari, L., Kerkemeier, S., Lan, Y.-H., Merzari, E., Min, M., Pazner, W., Ratnayaka, T., Shephard, M. S., Siboni, M.H., Smith, C.W., Thompson, J.L., Tomov, S., Warburton, T., 2021. ECP Milestone Report: High-order algorithmic developments and optimizations for large-scale GPU-accelerated simulations, WBS, Milestone CEED-MS36.

CEED MS35: link: Kolev, T., Fischer, P., Abdelfattah, A., Barra, V., Beams, N., Brown, J., Camier, J.S., Chalmers, N., Dobrev, V., Kerkemeier, S., Lan, Y.H., Merzari, E., Min, M., Phillips, M., Ratnayaka, T., Rowe, K., Thompson, J., Tomboulides, A., Tomov, S., Tomov, V,. and Warburton, T., 2020. ECP Milestone Report: Support CEED-enabled ECP applications in their preparation for Aurora/Frontier, WBS, Milestone CEED-MS35.

CEED MS34: link: Kolev, T., Fischer, P., Abdelfattah, A., Ananthan, S., Barra, V., Beams, N., Bleile, R., Brown, J., Carson, R,. Camier, J.S., Churchfield, M., Dobrev, V., Dongarra, J., Dudouit, Y., Karakus, A., Kerkemeier, S., Lan, Y., Medina, D., Merzari, E., Min, M., Parker, S., Ratnayaka, T., Smith, C., Sprague, M., Stitt, T., Thompson, J., Tomboulides, A., Tomov, S., Tomov, V., Vargas, A., Warburton, T., and Weiss, K., 2020. ECP Milestone Report: Improve performance and capabilities of CEED-enabled ECP applications on Summit/Sierra, WBS, Milestone CEED-MS34.

CEED MS29: link: Shephard, M., Barra, V., Brown, J., Camier, J.S., Dudouit, Y., Fischer, P., Kolev, T., Medina, D., Min, M., Smith, C., Siboni, M.H., Thompson, J., and Warburton, T., 2019. ECP Milestone Report: Improved Support for Parallel Adaptive Simulation in CEED, WBS, Milestone CEED-MS29.

CEED MS25: link: Brown, J., Abdelfattah, A., Barra, V., Dobrev, V., Dudouit, Y., Fischer, P., Kolev, T., Medina, D., Min, M., Ratnayaka, T., Smith, C., Thompson, J., Tomov, S., Tomov, V., and Warburton, T., 2019. ECP Milestone Report: Public release of CEED 2.0, WBS, Milestone CEED-MS25.

CEED MS23: link: Min, M., Camier, J.S., Fischer, P., Karakus, A., Kerkemeier, S., Kolev, T., Lan, Y.H., Medina, D., Merzari, E., Obabko, A., Ratnayaka, T., Shaver, D., Tomboulides, A., Tomov, V., and Warburton, T., 2018. ECP Milestone Report: Engage second wave ECP/CEED applications, WBS 2.2. 6.06, Milestone CEED-MS23.

CEED MS20: link: Tomov, S., Bello-Maldonado, P., Brown, J., Camier, J.S., Dobrev, V., Dongarra, J., Fischer, P., Haidar, A., Kolev, T., Merzari, E., Min, M., Obabko, A., Parker, S., Ratnayaka, T., Thompson, J., Abdelfattah, A., Tomov, V., and Warburton, T., 2018. ECP Milestone Report: Performance tuning of CEED software and first wave apps, WBS, Milestone CEED-MS20.