Robert C. Kirby

A compiler for variational forms

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 incompressible Navier--Stokes equations and linear elasticity. The speedup compared to the standard quadrature-based approach is impressive; in some cases the speedup is as large as a factor of 1000.

High Resolution Schemes for Conservation Laws with Locally Varying Time Steps

We develop upwind methods which use limited high resolution corrections in the spatial discretization and local time stepping for forward Euler and second order time discretizations.

Algorithm 839: FIAT, a new paradigm for computing finite element basis functions

L^\infty

Optimizing the Evaluation of Finite Element Matrices

stability is proven for both time stepping schemes for problems in one space dimension. These methods are restricted by a local CFL condition rather than the traditional global CFL condition, allowing local time refinement to be coupled with local spatial refinement. Numerical evidence demonstrates the stability and accuracy of the methods for problems in both one and two space dimensions.

Efficient compilation of a class of variational forms

Much of finite element computation is constrained by the difficulty of evaluating high-order nodal basis functions. While most codes rely on explicit formulae for these basis functions, we present a new approach that allows us to construct a general class of finite element basis functions from orthonormal polynomials and evaluate and differentiate them at any points. This approach relies on fundamental ideas from linear algebra and is implemented in Python using several object-oriented and functional programming techniques.

Efficient Assembly of

Assembling stiffness matrices represents a significant cost in many finite element computations. We address the question of optimizing the evaluation of these matrices. By finding redundant computations, we are able to significantly reduce the cost of building local stiffness matrices for the Laplace operator and for the trilinear form for Navier--Stokes operators. For the Laplace operator in two space dimensions, we have developed a heuristic graph algorithm that searches for such redundancies and generates code for computing the local stiffness matrices. Up to cubics, we are able to build the stiffness matrix on any triangle in less than one multiply-add pair per entry. Up to sixth degree, we can do it in less than about two pairs. Preliminary low-degree results for Poisson and Navier-Stokes operators in three dimensions are also promising.

H(\mathrm{div})

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 theorem, we design an algorithm for efficient pretabulation of the reference tensor. The new algorithm has been implemented in the FEniCS Form Compiler (FFC) and improves on a previous loop-based implementation by several orders of magnitude, thus shortening compile-times and development cycles for users of FFC.

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In this paper, we discuss how to efficiently evaluate and assemble general finite element variational forms on

H(\mathrm{curl})

H(\mathrm{div})

Conforming Finite Elements

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