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In this article, we introduce spyro, a software stack to solve wave propagation in heterogeneous domains and perform full waveform inversion (FWI) employing the finite-element framework from Firedrake, a high-level Python package for the automated solution of partial differential equations using the finite-element method. The capability of the software is demonstrated by using a continuous Galerkin approach to perform FWI for seismic velocity model building, considering realistic geophysics examples. A time domain FWI approach that uses meshes composed of variably sized triangular elements to discretize the domain is detailed. To resolve both the forward and adjoint-state equations and to calculate a mesh-independent gradient associated with the FWI process, a fully explicit, variable higher-order (up to degree k=5 in 2D and k=3 in 3D) mass-lumping method is used. We show that, by adapting the triangular elements to the expected peak source frequency and properties of the wave field (e.g., local P-wave speed) and by leveraging higher-order basis functions, the number of degrees of freedom necessary to discretize the domain can be reduced. Results from wave simulations and FWIs in both 2D and 3D highlight our developments and demonstrate the benefits and challenges with using triangular meshes adapted to the material properties.

We present an implementation of the trimmed serendipity finite element family, using the open source finite element package Firedrake. The new elements can be used seamlessly within the software suite for problems requiring $H^1$, \hcurl, or \hdiv-conforming elements on meshes of squares or cubes. To test how well trimmed serendipity elements perform in comparison to traditional tensor product elements, we perform a sequence of numerical experiments including the primal Poisson, mixed Poisson, and Maxwell cavity eigenvalue problems. Overall, we find that the trimmed serendipity elements converge, as expected, at the same rate as the respective tensor product elements while being able to offer significant savings in the time or memory required to solve certain problems., 19 pages, 7 figures, 3 tables, 2 listings

Numerical resolution of exterior Helmholtz problems requires some approach to domain truncation. As an alternative to approximate nonreflecting boundary conditions and invocation of the Dirichlet-to-Neumann map, we introduce a new, nonlocal boundary condition. This condition is exact and requires the evaluation of layer potentials involving the free space Green's function. However, it seems to work in general unstructured geometry, and Galerkin finite element discretization leads to convergence under the usual mesh constraints imposed by G{\aa}rding-type inequalities. The nonlocal boundary conditions are readily approximated by fast multipole methods, and the resulting linear system can be preconditioned by the purely local operator involving transmission boundary conditions.

The Morse–Ingard equations of thermoacoustics Morse and Ingard (1986) are a system of coupled time-harmonic equations for the temperature and pressure of an excited gas. They form a critical aspect of modeling trace gas sensors. In this paper, we analyze a reformulation of the system that has a weaker coupling between the equations than the original form. We give a Garding-type inequality for the system that leads to optimal-order asymptotic finite element error estimates. We also develop preconditioners for the coupled system. These are derived by writing the system as a 2 × 2 block system with pressure and temperature unknowns segregated into separate blocks and then using either the block diagonal or block lower triangular part of this matrix as a preconditioner. Consequently, the preconditioner requires inverting smaller, Helmholtz-like systems individually for the pressure and temperature. Rigorous eigenvalue bounds are given for the preconditioned system, and these are supported by numerical experiments.

Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolat...

In this article, we introduce spyro, a software stack to solve acoustic wave propagation in heterogeneous domains and perform full waveform inversion (FWI) employing the finite element framework from Firedrake, a high-level Python package for the automated solution of partial differential equations using the finite element method. The capability of the software is demonstrated by using a continuous Galerkin approach to perform FWI for seismic velocity model building, considering realistic geophysics examples. A time-domain FWI approach is detailed that uses meshes composed of variably sized triangular elements to discretize the domain. To resolve both the forward and adjoint-state equations, and to calculate a mesh-independent gradient associated with the FWI process, a fully-explicit, variable higher-order (up to degree k = 5 in 2D and k = 3 in 3D) mass lumping method is used. We show that, by adapting the triangular elements to the expected peak source frequency and properties of the wavefield (e.g., local P-wavespeed) and by leveraging higher-order basis functions, the number of degrees-of-freedom necessary to discretize the domain can be reduced. Results from wave simulations and FWIs in both 2D and 3D highlight our developments and demonstrate the benefits and challenges with using triangular meshes adapted to the material properties.

The finite element method (FEM) is a standard tool for solving differential equations in many disciplines, e.g., electromagnetics, solid and structural mechanics, fluid dynamics, acoustics, and thermal conduction. Jin [40, 41] and Peterson [54] give good accounts of the FEM for electromagnetics. More mathematical treatments of the same topic are given in [12, 48]. This chapter gives an introduction to FEM in general and FEM for Maxwell’s equations in particular. Practical issues, such as how to handle unstructured grids and how to write FEM programs, will be discussed in some detail.

While implicit Runge–Kutta (RK) methods possess high order accuracy and important stability properties, implementation difficulties and the high expense of solving the coupled algebraic system at each time step are frequently cited as impediments. We present Irksome , a high-level library for manipulating UFL (Unified Form Language) expressions of semidiscrete variational forms to obtain UFL expressions for the coupled Runge–Kutta stage equations at each time step. Irksome works with the Firedrake package to enable the efficient solution of the resulting coupled algebraic systems. Numerical examples confirm the efficacy of the software and our solver techniques for various problems.

We describe a fully discrete mixed finite element method for the linearized rotating shallow water model, possibly with damping. While Crank–Nicolson time-stepping conserves energy in the absence of drag or forcing terms and is not subject to a CFL-like stability condition, it requires the inversion of a linear system at each step. We develop weighted-norm preconditioners for this algebraic system that are nearly robust with respect to the physical and discretization parameters in the system. Numerical experiments using Firedrake support the theoretical results.

Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to interesting linear algebra questions. When attempting to find a polynomial approximation of boundary or initial data, one encounters the Bernstein-Vandermonde matrix, which is found to be highly ill-conditioned. Previously, we used the relationship between monomial Bezout matrices and the inverse of Hankel matrices to obtain a decomposition of the inverse of the Bernstein mass matrix in terms of Hankel, Toeplitz, and diagonal matrices. In this paper, we use properties of the Bernstein-Bezout matrix to factor the inverse of the Bernstein-Vandermonde matrix into a difference of products of Hankel, Toeplitz, and diagonal matrices. We also use a nonstandard matrix norm to study the conditioning of the Bernstein-Vandermonde matrix, showing that the conditioning in this case is better than in the standard 2-norm. Additionally, we use properties of multivariate Bernstein polynomials to derive a block $LU$ decomposition of the Bernstein-Vandermonde matrix corresponding to equispaced nodes on the $d$-simplex., Comment: 21 pages, 4 figures

We consider the mass matrix arising from Bernstein polynomials as finite element shape functions. In particular, we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the Bernstein representation of orthogonal polynomials. We then derive a fast algorithm for solving linear systems involving the element mass matrix. After establishing these results, we describe the application of Bernstein techniques to the discontinuous Galerkin finite element method for hyperbolic conservation laws, obtaining optimal complexity algorithms. Finally, we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run-time for the function evaluation needed in DG time-stepping.

We study mixed finite element methods for the rotating shallow water equations with linearized momentum terms but nonlinear drag. By means of an equivalent second-order formulation, we prove long-time stability of the system without energy accumulation. We also give rates of damping in unforced systems and various continuous dependence results on initial conditions and forcing terms. A priori error estimates for the momentum and free surface elevation are given in $$L^2$$ as well as for the time derivative and divergence of the momentum. Numerical results confirm the theoretical results regarding both energy damping and convergence rates.

Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite element transformations in FInAT and hence into the Firedrake finite element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly-enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using $C^1$ discretizations., Comment: 23 pages

We consider a time-harmonic mathematical model for a coupled thermal-acoustic model. We demonstrate coercivity of the PDE system and hence well-posedness and optimal error estimates for standard Galerkin methods. The large scale and ill-conditioning of the discrete system motivates the study of preconditioners. Three block preconditioners are studied. The first two, applying block Jacobi and block Gauss-Seidel approaches to our problem, give provably mesh-independent spectral bounds. The third preconditioner, based on problem-specific physical reasoning, does not have this property. However, on moderate-sized meshes, it considerably outperforms the existing mesh techniques. Numerical experiments in two and three dimensions confirm the theoretical results.

The use of a reference element on which a finite element basis is constructed once and mapped to each cell in a mesh greatly expedites the structure and efficiency of finite element codes. However, many famous finite elements such as Hermite, Morley, Argyris, and Bell, do not possess the kind of equivalence needed to work with a reference element in the standard way. This paper gives a generalizated approach to mapping bases for such finite elements by means of studying relationships between the finite element nodes under push-forward., Comment: 28 pages

We apply mixed finite element approximations to the first-order form of the acoustic wave equation. The semidiscrete method exactly conserves the system energy. A fully discrete method employing the symplectic Euler time method in time exactly conserves a positive-definite pertubed energy functional that is equivalent to the actual energy under a CFL condition. In addition to proving optimal-order $$L^\infty (L^2)$$L?(L2) estimates, we also develop a bootstrap technique that allows us to derive stability and error bounds for the time derivatives and divergence of the vector variable beyond the standard under some additional regularity assumptions.

We revisit the question of optimizing the construction and application of finite element matrices. By using commuting properties of the reference mappings and duality, we reorganize stiffness matrix construction and matrix-free application so that the bulk of the work can be done by optimized matrix multiplication libraries. We provide examples, including numerical experiments, with the Laplace and curl-curl operators as well as develop a general framework. Our techniques are applicable in general geometry and are not restricted to constant coefficient operators.

We combine recently developed finite element algorithms based on Bernstein polynomials [M. Ainsworth, G. Andriamaro, and O. Davydov, SIAM J. Sci. Comput., 33 (2011), pp. 3087--3109; R. C. Kirby, Numer. Math., 117 (2011), pp. 631--652] with the explicit basis construction of the finite element exterior calculus [D. N. Arnold, R. S. Falk, and R. Winther, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1660--1672] to give a family of algorithms for the de Rham complex on simplices that achieves stiffness matrix construction and matrix-free action in optimal complexity. These algorithms are based on realizing the exterior calculus bases as short combinations of Bernstein polynomials. Numerical results confirm optimal scaling of the algorithms and favorable comparison with FEniCS at high polynomial order as well. Additional empirical studies show that very high accuracy is achieved in the mixed discretization of the Poisson equation and the Maxwell eigenvalue problem as the polynomial degree increases.

Finite element discretizations of multiphysics problems frequently give rise to block-structured linear algebra problems that require effective preconditioners. We build two classes of preconditioners in the spirit of well-known block factorizations [M. F. Murphy, G. H. Golub, and A. J. Wathen, SIAM J. Sci. Comput., 21 (2000), pp. 1969--1972; I. C. F. Ipsen, SIAM J. Sci. Comput., 23 (2001), pp. 1050--1051] and apply these to the diffusive portion of the bidomain equations and the Benard convection problem. An abstract generalized eigenvalue problem allows us to give application-specific bounds for the real parts of eigenvalues for these two problems. This analysis is accompanied by numerical calculations with several interesting features. One of our preconditioners for the bidomain equations converges in five iterations for a range of problem sizes. For Benard convection, we observe mesh-independent convergence with reasonable robustness with respect to physical parameters, and offer some preliminary para...

A domain-specific embedded language called ibvp was developed to model initial-boundary value problems for partial differential equations. As a (motivating) application, we developed tools to parse this language and generate problem-specific input to the Proteus toolkit developed at ERDC.

SUMMARYWe derive block preconditioners for a finite element discretization of incompressible flow coupled toheat transport by the Boussinesq approximation. Our techniques rely on effectively approximating theSchur complement obtained by eliminating the fluid variables to obtain an equation for temperature alone.Additionally, the method utilizes existing block-structured preconditioners and multilevel methods for theNavier—Stokes equations and scalar convection-diffusion. We find that the preconditioner remains robustand scalable even when the subsolves are applied quite inexactly. Copyright c 0000 John Wiley & Sons,Ltd. Received ... KEY WORDS: Finite element, incompressible flow, Benard convection, multiphysics, block precondi-´tioner. 1. INTRODUCTIONIn this paper, we extend block preconditioners for inf-sup stable finite element discretizations of theincompressible Navier–Stokes equations, u+ uru+ rp= fru= 0;(1)posed on some domain ˆR d for d= 2;3 and equipped with appropriate boundary conditions, toa particular non-dimensionalization of a coupled fluid-thermal problem, Benard convection:´ u+ uru+ rp= RaPrgT^ru= 01Pr T+ urT= 0;(2)again posed on some domain along with boundary conditions. The fluid velocity and pressureare uand p, respectively, in both equations, and the temperature is T. The in (1) is the fluidviscosity. The Rayleigh number Rameasures the ratio of energy from buoyant forces to viscous

Intrepid is a Trilinos package for advanced discretizations of Partial Differential Equations (PDEs). The package provides a comprehensive set of tools for local, cell-based construction of a wide range of numerical methods for PDEs. This paper describes the mathematical ideas and software design principles incorporated in the package. We also provide representative examples showcasing the use of Intrepid both in the context of numerical PDEs and the more general context of data analysis.

We derive low-complexity matrix-free finite element algorithms for simplicial Bernstein polynomials on simplices. Our techniques, based on a sparse representation of differentiation and special block structure in the matrices evaluating B-form polynomials at warped Gauss points, apply to variable coefficient problems as well as constant coefficient ones, thus extending our results in Kirby (Numer Math, 2011, in press).

Fast algorithms for applying finite element mass and stiffness operators to the B-form of polynomials over d-dimensional simplices are derived. These rely on special properties of the Bernstein basis and lead to stiffness matrix algorithms with the same asymptotic complexity as tensor-product techniques in rectangular domains. First, special structure leading to fast application of mass matrices is developed. Then, by factoring stiffness matrices into products of sparse derivative matrices with mass matrices, fast algorithms are also obtained for stiffness matrices.

In this paper, we discuss how to efficiently evaluate and assemble general finite element variational forms on H(div) and H(curl). The proposed strategy relies on a decomposition of the element tensor into a precomputable reference tensor and a mesh-dependent geometry tensor. Two key points must then be considered: the appropriate mapping of basis functions from a reference element, and the orientation of geometrical entities. To address these issues, we extend here a previously presented representation theorem for affinely mapped elements to Piola-mapped elements. We also discuss a simple numbering strategy that removes the need to contend with directions of facet normals and tangents. The result is an automated, efficient, and easy-to-use implementation that allows a user to specify finite element variational forms on H(div) and H(curl) in close to mathematical notation.

The L 2 -orthogonal polynomials used in finite and spectral element methods on nonrectangular elements may be defined in terms of collapsed coordinates, wherein the shapes are mapped to a square or cube by means of a singular change of variables. The orthogonal basis is a product of specific Jacobi polynomials in these new coordinates. Implementations of these polynomials require special handling of the coordinate singularities. We derive new recurrence relations for these polynomials on triangles and tetrahedra that work directly in the original coordinates. These relations, also applicable to pyramids and prisms, do not require any special treatment of singular points. These recurrences are seen to speed up both symbolic and numerical computation of the orthogonal polynomials.

We examine condition numbers, preconditioners, and iterative methods for finite element discretizations of coercive PDEs in the context of the fundamental solvability result, the Lax-Milgram lemma. Working in this Hilbert space context is justified because finite element operators are restrictions of infinite-dimensional Hilbert space operators to finite-dimensional subspaces. Moreover, useful insight is gained as to the relationship between Hilbert space and matrix condition numbers, and translating Hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes. In this framework, the “simplest” preconditioner for an operator from a Hilbert space into its dual is the Riesz isomorphism. Simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen. Moreover, the abstraction allows us to consider not only Riesz map preconditioning for convection-diffusion equations in $H^1$ but also operators on other Hilbert spaces, such as planar elasticity in $\left(H^1\right)^2$.

Computational analysis of systems governed by partial differential equations (PDEs) requires not only the calculation of a solution but the extraction of additional information such as the sensitivity of that solution with respect to input parameters or the inversion of the system in an optimization or design loop. Moving beyond the automation of discretization of PDEs by finite element methods, we present a mathematical framework that unifies the discretization of PDEs with these additional analysis requirements. In particular, Frechet differentiation on a class of functionals together with a high-performance finite element framework has led to a code, called Sundance, that provides high-level programming abstractions for the automatic, efficient evaluation of finite variational forms together with the derived operators required by engineering analysis.

This paper continues earlier work on mathematical techniques for generating optimized algorithms for computing finite element stiffness matrices. These techniques start from representing the stiffness matrix for an affine element as a collection of contractions between reference tensors and an element-dependent geometry tensor. We go beyond the complexity-reducing binary relations explored in [R. C. Kirby, A. Logg, L. R. Scott, and A. R. Terrel, SIAM J. Sci. Comput., 28 (2006), pp. 224-240] to consider geometric relationships between three or more objects. Algorithms based on these relationships often have even fewer operations than those based on complexity-reducing relations.

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 1000., 27 pages, 10 figures

Our previous work on FIAT (Finite Element Automatic Tabulator) developed a “computational representation theory ” that allowed us to construct arbitrary order instances of a wide range of finite elements, many of which are infrequently used owing to their associated code complexity. In our present work, we further hone this theory by rephrasing most of the internal operations as linear transformations over finite-dimensional Banach spaces. This additional insight has led to increased code granularity and allowed the use of level 3 BLAS operations. This is both a conceptual and a practical development; as the run-time performance of FIAT has been improved multiple orders of magnitude.

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.

We reinterpret the expression templates for matrix computation put forward by other authors in light of functional programming ideas. Existing expression template libraries frequently focus on componentwise operations such as addition at the expense of matrix multiplication. The ideas presented are implemented in a C++ library called LLANO (Lazy Linear Algebraic Numerical Objects).

In this paper, residual-based a posteriori error bounds are derived for the mixed finite element method applied to a model second order elliptic problem. A global upper bound for the error in the scalar variable is established, as well as a local lower bound. In addition, due to the fact that the scalar and vector variables are approximated to equal order accuracy, the dual problem may be modified to give an upper bound for the vector variable. Some comments on estimating more general error quantities are also made. The estimate effectively guides adaptive refinement for a smooth problem with a boundary layer, as well as detects the need to refine near a singularity.

A class of finite volume methods based on standard high resolution schemes, but which allows spatially varying time steps, is described and analyzed. A maximum principle and the TVD property are verified for general advective flux, extending the previous theoretical work on local time stepping methods. Moreover, an entropy condition is verified which, with sufficient limiting, guarantees convergence to the entropy solution for convex flux.

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. $L^\infty$ 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.

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., ACM Transactions on Mathematical Software 33(3), 20 pages (2007)

We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. 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 present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization.

We examine the effect of using complexity-reducing relations to generate optimized code for the evaluation of finite element variational forms. The optimizations are implemented in a prototype code named FErari, which has been integrated as an optimizing backend to the FEniCS Form Compiler, FFC. In some cases, FErari provides very little speedup, while in other cases, we obtain reduced local operation counts of a factor of as much as 7.9 and speedups for the assembly of the global sparse matrix of as much as a factor of 2.8., ACM Transactions on Mathematical Software 35(2), 18 pages (2008)

This paper introduces Playa, a high-level user interface layer for composing algorithms for complex multiphysics problems out of objects from other Trilinos packages. Among other features, Playa provides very high-performance overloaded operators implemented through an expression template mechanism. In this paper, we give an overview of the central Playa objects from a user's perspective, show application to a sequence of increasingly complex solver algorithms, provide timing results for Playa's overloaded operators and other functions, and briefly survey some of the implementation issues involved.

This chapter describes the mathematical framework for constructing a general class of finite elements on reference domains. This framework is used by both the FIAT and SyFi projects, see the Chapters 13 and 15, respectively. Our goal is to provide a framework by which simplicial finite elements with very complicated bases can be constructed automatically. We work from the classic Ciarlet definition of the finite element and its “nodal” basis (an abstract notion far more general and powerful than standard node-oriented Lagrange polynomials).

This chapter provides a glimpse of the considerable range of finite elements in the literature. Many of the elements presented here are implemented as part of the FEniCS project already; some are future work. The universe of finite elements extends far beyond what we consider here. In particular, we consider only simplicial, polynomial-based elements. We thus bypass elements defined on quadrilaterals and hexahedra, composite and macro-element techniques, as well as XFEM-type methods. Even among polynomial-based elements on simplices, the list of elements can be extended. Nonetheless, this chapter presents a comprehensive collection of some the most common, and some more unusual, finite elements.

In Chapter 6, we saw that an important step in the assembly of matrices and vectors for the 4523 discretization of finite element variational problems is the evaluation of the cell (element) tensor AT 4524 defined by 4525 AT,i = aT(f T,r ir,...,fT,2 i2,fT,1 i1 ).

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. 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. Preliminary low-degree results for Poisson and Navier-Stokes operators in three dimensions are also promising.

The tensor contraction structure for the computation of the element tensor AT obtained in Chapter 8,enables not only the construction of a compiler for variational forms,but an optimizing compiler.For typical variational forms,the reference tensor A0 has significant structure that allows the element tensor AT to be computed on an arbitrary cell T at a lower computational cost.

In Chapter 8, we presented a framework for efficient evaluation of multilinear forms based on 7311 expressing the multilinear form as a special tensor contraction. This allows generation of efficient 7312 low-level code for assembly of a range of multilinear forms.

Much of the FEniCS software is devoted to the formulation of variational forms (UFL), the discretization of variational forms (FIAT, FFC, SyFi) and the assembly of the corresponding discrete operators (UFC, DOLFIN). This chapter summarizes the notation for variational forms used throughout FEniCS.