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💎 Feel++: Finite Element Embedded Language and Library in C++


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Feel++: Finite Element Embedded Library in C++

feelpp DOI

feelpp?color=009688&logo=Riseup&style=flat square feelpp?color=009688&logo=Moleculer&logoColor=white&style=flat square feelpp?color=009688&logo=Bilibili&logoColor=white&style=flat square feelpp?logo=Draugiem feelpp?color=009688&style=flat square&logo=Hack The Box&logoColor=white

Feel++ is a C++ library for continuous or discontinuous Galerkin methods including finite element method(FEM), spectral element methods(SEM), reduced basis methods, discontinuous galerkin methods (DG and HDG) in 1D 2D and 3D and in parallel. Checkout What is Feel++?, Feel++ Features and some Examples.


The latest release of Feel++ is here feelpp

Feel++ has a DOI provided by Zenodo. Please use this to cite Feel++ if you use it in a publication DOI

Feel++ is split into three components:


library and tools


mono and multiphysics toolboxes (cfd, csm, heat transfer, fsi, heat and fluid, hdg(poisson and elasticity), thermo-electric and maxwell)


model order reduction applications and tools

These components are built and delivered in two distribution channels: stable and latest. The channels are currently available via Docker containers, Debian and Ubuntu packages.


Once a year, sometimes more, we make a release of Feel++ and it becomes the basis of the stable channel. The channel is updated infrequently, only for a new release or a major bug.


Feel++ has a very active development and changes are made everyday with the research done by Cemosis and its collaborators. Each commit in the main development branch triggers a new full build with more than 800 tests from unit test to full pde solves.

Instructions are available here to install Feel++ :

Feel++ Documentation

Slack Discussion Forum

We encourage you to ask questions and discuss any aspects of the project on Slack. New contributors are always welcome!

Continuous Integration

Feel++ maintains various branches. At the core, the development model is greatly inspired by existing models out there. The central repo holds two main branches with an infinite lifetime: master and develop


Main branch where the source code of HEAD always reflects a production-ready state.


Main branch where the source code of HEAD always reflects a state with the latest delivered development changes for the next release. Some would call this the “integration branch”. This is where any automatic nightly builds are built from.


Feature branches (or sometimes called topic branches) are used to develop new features for the upcoming or a distant future release. When starting development of a feature, the target release in which this feature will be incorporated may well be unknown at that point. The essence of a feature branch is that it exists as long as the feature is in development, but will eventually be merged back into develop (to definitely add the new feature to the upcoming release) or discarded (in case of a disappointing experiment).

What is Feel++?

Feel++ is a C++ library for continuous or discontinuous Galerkin methods including finite element method(FEM), spectral element methods(SEM), reduced basis methods, discontinuous Galerkin methods (DG and HDG) in 1D 2D and 3D and in parallel. The objectives of this framework are quite ambitious; ambitions which could be expressed in various ways such as :

  • the creation of a versatile mathematical kernel solving easily problems using different techniques thus allowing testing and comparing methods, e.g. cG versus dG,

  • the creation of a small and manageable library which shall nevertheless encompass a wide range of numerical methods and techniques,

  • build mathematical software that follows closely the mathematical abstractions associated with partial differential equations (PDE),

  • the creation of a library entirely in C++ allowing to create complex and typically multi-physics applications such as fluid-structure interaction or mass transport in haemodynamic.


  • 1D 2D and 3D (including high order) geometries and also lower topological dimension 1D(curve) in 2D and 3D or 2D(surface) in 3D

  • continuous and discontinuous (dG and hdG) arbitrary order Galerkin Methods in 1D, 2D and 3D including finite and spectral element methods

  • domain specific embedded language in C++ for variational formulations

  • interfaced with PETSc for linear and non-linear solvers

  • seamless parallel computations using PETSc

  • interfaced with SLEPc for large-scale sparse standard and generalized eigenvalue solvers

  • supports Gmsh for mesh generation

  • supports Gmsh for post-processing (including on high order geometries)

  • supports Paraview and CEI/Ensight for post-processing and the following file formats: ensight gold, gmsh, xdmf.


In the spirit of free software, everyone is encouraged to help improve this project. If you discover errors or omissions in the source code, documentation, or website content, please don’t hesitate to submit an issue or open a pull request with a fix. New contributors are always welcome!

Here are some ways you can contribute:

  • by using develop versions

  • by reporting bugs

  • by suggesting new features

  • by writing or editing documentation

  • by writing specifications

  • by writing code — No patch is too small.

    • fix typos

    • add comments

    • write examples!

    • write tests!

  • by refactoring code

  • by fixing issues

  • by reviewing Pull Requests

The Contributing guide provides information on how to create, style, and submit issues, feature requests, code, and documentation to the Feel++ Project.

Getting Help

The Feel++ project is developed to help you easily do (i) modelisation simulation and optimisation and (ii) high performance computing. But we can’t do it without your feedback! We encourage you to ask questions and discuss any aspects of the project on the discussion list, on Twitter or in the chat room.


#feelpp hashtag or @feelpp mention

Chat (Slack)


Further information and documentation about Feel++ can be found on the project’s website.

Home | News | Docs

The Feel++ organization on GitHub hosts the project’s source code, issue tracker, and sub-projects.

Source repository (git)

Issue tracker

Feel++ organization on GitHub

Copyright © 2011-2023 Feel++ Consortium. Free use of this software is granted under the terms of the GPL License.

See the LICENSE file for details.


Feel++ is led by Christophe Prud’homme and has received contributions from many other individuals.


Laplacian in 2D using P3 Lagrange basis functions

Here is a full example to solve

-\Delta u = f \mbox{ in } \Omega,\quad u=g \mbox{ on } \partial \Omega

#include <feel/feel.hpp>

int main(int argc, char**argv )
    using namespace Feel;
    Environment env( _argc=argc, _argv=argv,
                                  _author="Feel++ Consortium",

    auto mesh = unitSquare();
    auto Vh = Pch<1>( mesh );
    auto u = Vh->element();
    auto v = Vh->element();

    auto l = form1( _test=Vh );
    l = integrate(_range=elements(mesh),

    auto a = form2( _trial=Vh, _test=Vh );
    a = integrate(_range=elements(mesh),
                  _expr=gradt(u)*trans(grad(v)) );
    a+=on(_range=boundaryfaces(mesh), _rhs=l, _element=u,
          _expr=constant(0.) );

    auto e = exporter( _mesh=mesh, _name="qs_laplacian" );
    e->add( "u", u );
    return 0;

Bratu equation in 2D

Here is a full non-linear example - the Bratu equation - to solve

\[-\Delta u + e^u = 0 \mbox{ in } \Omega,\quad u=0 \mbox{ on } \partial \Omega$$.\]
#include <feel/feel.hpp>

    Feel::po::options_description bratuoptions( "Bratu problem options" );
    ( "lambda", Feel::po::value<double>()->default_value( 1 ),
                "exp() coefficient value for the Bratu problem" )
    ( "penalbc", Feel::po::value<double>()->default_value( 30 ),
                 "penalisation parameter for the weak boundary conditions" )
    ( "hsize", Feel::po::value<double>()->default_value( 0.1 ),
               "first h value to start convergence" )
    ( "export-matlab", "export matrix and vectors in matlab" )
    return bratuoptions.add( Feel::feel_options() );

 * Bratu Problem
 * solve \f$ -\Delta u + \lambda \exp(u) = 0, \quad u_\Gamma = 0\f$ on \f$\Omega\f$
main( int argc, char** argv )

    using namespace Feel;
    Environment env( _argc=argc, _argv=argv,
                                  _author="Christophe Prud'homme",
    auto mesh = unitSquare();
    auto Vh = Pch<3>( mesh );
    auto u = Vh->element();
    auto v = Vh->element();
    double penalbc = option(_name="penalbc").as<double>();
    double lambda = option(_name="lambda").as<double>();

    auto Jacobian = [=](const vector_ptrtype& X, sparse_matrix_ptrtype& J)
            auto a = form2( _test=Vh, _trial=Vh, _matrix=J );
            a = integrate( elements( mesh ), gradt( u )*trans( grad( v ) ) );
            a += integrate( elements( mesh ), lambda*( exp( idv( u ) ) )*idt( u )*id( v ) );
            a += integrate( boundaryfaces( mesh ),
               ( - trans( id( v ) )*( gradt( u )*N() ) - trans( idt( u ) )*( grad( v )*N()  + penalbc*trans( idt( u ) )*id( v )/hFace() ) );
    auto Residual = [=](const vector_ptrtype& X, vector_ptrtype& R)
            auto u = Vh->element();
            u = *X;
            auto r = form1( _test=Vh, _vector=R );
            r = integrate( elements( mesh ), gradv( u )*trans( grad( v ) ) );
            r +=  integrate( elements( mesh ),  lambda*exp( idv( u ) )*id( v ) );
            r +=  integrate( boundaryfaces( mesh ),
               ( - trans( id( v ) )*( gradv( u )*N() ) - trans( idv( u ) )*( grad( v )*N() ) + penalbc*trans( idv( u ) )*id( v )/hFace() ) );
    backend()->nlSolver()->residual = Residual;
    backend()->nlSolver()->jacobian = Jacobian;
    backend()->nlSolve( _solution=u );

    auto e = exporter( _mesh=mesh );
    e->add( "u", u );