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We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which… (More)

We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies for… (More)

As a key step towards a complete automation of the finite element method, we present a new algorithm for automatic and efficient evaluation of multilinear variational forms. The algorithm has been implemented in the form of a compiler, the FEniCS Form Compiler (FFC). We present benchmark results for a series of standard variational forms, including the… (More)

Continuing the discussion of the multi-adaptive Galerkin methods mcG(q) and mdG(q) presented in [, we present adaptive algorithms for global error control, iterative solution methods for the discrete equations, features of the implementation Tanganyika, and computational results for a variety of ODEs. Examples include the Lorenz system, the solar system,… (More)

We present multi-adaptive versions of the standard continuous and discontinuous Galerkin methods for ODEs. Taking adaptivity one step further, we allow for individual time-steps, order and quadrature, so that in particular each individual component has its own time-step sequence. This paper contains a description of the methods, an analysis of their basic… (More)

- Anders Logg, Johan Hoffman, Johan Jansson, Claes Johnson, Robert C. Kirby, Matthew Knepley +1 other
- 2005

The finite element method can be viewed as a machine that automates the discretization of differential equations , taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations. However, the generality of the framework provided by the finite element method is seldom reflected in implementations… (More)

We investigate the compilation of general multilinear variational forms over affines simplices and prove a representation theorem for the representation of the element tensor (element stiffness matrix) as the contraction of a constant reference tensor and a geometry tensor that accounts for geometry and variable coefficients. Based on this representation… (More)

A compiler approach for generating low-level computer code from high-level input for discontinuous Galerkin finite element forms is presented. The input language mirrors conventional mathematical notation, and the compiler generates efficient code in a standard programming language. This facilitates the rapid generation of efficient code for general… (More)

We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then… (More)

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… (More)