Feel++ : A computational framework for Galerkin Methods and Advanced Numerical Methods

Abstract

Abstract. This paper presents an overview of a unified framework for finite element and spectral element methods in 1D, 2D and 3D in C ++ called FEEL++. The article is divided in two parts. The first part provides a digression through the design of the library as well as the main abstractions handled by it, namely, meshes, function spaces, operators, linear and bilinear forms and an embedded variational language. In every case, the closeness between the language developed in FEEL++ and the equivalent mathematical objects is highlighted. In the second part, examples using the mortar, Schwartz (non)overlapping, three fields and two fictitious domain-like methods (the Fat Boundary Method and the Penalty Method) are presented and numerically solved in the scope of the library.

Boundary conditions involving pressure for the Stokes problem and applications in computational hemodynamics

“…Feel++ provides a mathematical kernel for solving partial di erential equation using arbitrary order Galerkin methods ( , , , , ) in 1 , 2 , 3 and on manifolds using simplices and hypercubes meshes [31] : (i) a polynomial library allowing for a wide range polynomial expansions including H div and H curl elements, (ii) a light interface to B .UB , E 3 and PETS [3]/SLEP as well as a scalable in-house solution strategy (iii) a language for Galerkin methods starting with fundamental concepts such as function spaces, forms, operators, functionals and integrals, (iv) a framework that allows user codes to scale seamlessly from single core computation to thousands of cores and enables hybrid computing.…”

A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes

“…The framework of Sect. 2 serves as a basis for a DSEL targeted at expressing linear and bilinear forms using a syntax closely inspired by that of Feel++ [34,35]. To illustrate the capabilities of the DSEL in a nutshell, compare Listing 1 with the expression of the bilinear form a sip h (2.16).…”

“…Finite element libraries have nowadays reached a good level of maturity. Just to mention a few, we recall Feel++ [33][34][35], FEniCS [31], FreeFEM++ [29]. All of the above projects provide a userfriendly front-end in the form of a Domain Specific Language (DSL) possibly embedded in a general purpose, high-level hosting language (Domain Specific Embedded Language or DSEL).…”

Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

“…All operators related to the domain decomposition can be easily generated using finite element Domain-Specific Languages. We will be using FreeFem++ [62] since it has already been proven that it can enable large-scale simulations using overlapping Schwarz methods [63], but our framework interacts with other DSLs such as Feel++ [64].…”