Unified framework for finite element assembly

Over the last fifty years, the finite element method has emerged as a successful methodology for solving a wide range of partial differential equations. At the heart of any finite element simulation is the assembly of matrices and vectors from finite element variational forms. In this paper, we present a general and unified framework for finite element assembly. Based on this framework, we propose a specific software interface (UFC) between problem-specific and general-purpose components of finite element programs. The interface is general in the sense that it applies to a wide range of finite element problems (including mixed finite elements and discontinuous Galerkin methods) and may be used with libraries that differ widely in their design. The interface consists of a minimal set of abstract C++ classes and data transfer is via plain C arrays. We discuss how one may use the UFC interface to build a plug-and-play system for finite element simulation where basic components such as computational meshes, linear algebra and, in particular, variational form evaluation may come from different libraries and be used interchangeably. We further discuss how the UFC interface is used to glue together components from the FEniCS suite of software to provide an integrated highlevel environment where variational forms may be entered as expressions directly in Python and assembled efficiently into sparse matrices. A central design goal for the interface is to minimize dependency on external libraries for the problem-specific code used in applications. Thus, the UFC interface consists of a single C++ header file and does not rely on external libraries for its operation. In particular, the UFC interface does not depend on any other FEniCS components. As a result, finite element code developers may use the interface to detach equation specific details from general-purpose library code, allowing very flexible connections to alternative libraries. We encourage developers of finite element libraries to incorporate the interface in their libraries. The UFC interface is released into the public domain.