Advances in Feel++ : A Domain Specific Embedded Language in C++ for Partial Differential Equations

We present our advances in developing a language specific to partial differential equations embedded in C++. We have been developing the Feel++ framework (Finite Element method Embedded Language in C++) to the point where it allows to use a very wide range of Galerkin methods and advanced numerical methods such as domain decomposition methods including mortar and three fields methods, fictitious domain methods or certified reduced basis. We shall present an overview of the various ingredients as well as some illustrations. The ingredients include a very expressive embedded language, seamless interpolation, mesh adaption, seamless parallelisation and automatic differentiation using Frechet derivative. As to the illustrations, they exercise the versatility of the framework either by allowing the developement and/or numerical verification of (new) mathematical methods or the development of large multi-physics applications -- e.g. fluid-structure interaction using either an Arbitrary Lagrangian Eulerian formulation or a levelset based one; high field magnets modeling which involves electro-thermal, magnetostatics, mechanical and thermo-hydraulics model.