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Automating the Finite Element Method

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Abstract

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 (realizations), which are often specialized and can handle only a small set of variational problems and finite elements (but are typically parametrized over the choice of mesh).

This paper reviews ongoing research in the direction of a complete automation of the finite element method. In particular, this work discusses algorithms for the efficient and automatic computation of a system of discrete equations from a given variational problem, finite element and mesh. It is demonstrated that by automatically generating and compiling efficient low-level code, it is possible to parametrize a finite element code over variational problem and finite element in addition to the mesh.

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Abbreviations

A :

The differential operator of the model A(u)=f

A :

The global tensor with entries {A i }i∈ℐ

A 0 :

The reference tensor with entries \(\{A^{0}_{i\alpha}\}_{i\in\mathcal{I}_{K},\alpha\in\mathcal{A}}\)

\(\bar{A}^{0}\) :

The matrix representation of the (flattened) reference tensor A 0

A K :

The element tensor with entries \(\{A^{K}_{i}\}_{i\in\mathcal{I}_{K}}\)

a :

The semilinear, multilinear or bilinear form

a K :

The local contribution to a multilinear form a from K

a K :

The vector representation of the (flattened) element tensor A K

\(\mathcal{A}\) :

The set of secondary indices

ℬ:

The set of auxiliary indices

e :

The error, e=U−u

F K :

The mapping from K 0 to K

G K :

The geometry tensor with entries \(\{G_{K}^{\alpha}\}_{\alpha\in\mathcal{A}}\)

g K :

The vector representation of the (flattened) geometry tensor G K

ℐ:

The set ∏ rj=1 [1,N j] of indices for the global tensor A

ℐ K :

The set ∏ rj=1 [1,n j K ] of indices for the element tensor A K (primary indices)

ι K :

The local-to-global mapping from \(\mathcal{N}_{K}\) to \(\mathcal{N}\)

\(\hat{\iota}_{K}\) :

The local-to-global mapping from \(\hat{\mathcal{N}}_{K}\) to \(\hat{\mathcal{N}}\)

ι j K :

The local-to-global mapping from \(\mathcal{N}_{K}^{j}\) to \(\mathcal{N}^{j}\)

K :

A cell in the mesh  \(\mathcal{T}\)

K 0 :

The reference cell

L :

The linear form (functional) on \(\hat{V}\) or \(\hat{V}_{h}\)

m :

The number of discrete function spaces used in the definition of a

N :

The dimension of \(\hat{V}_{h}\) and V h

N j :

The dimension V j h

N q :

The number of quadrature points on a cell

n 0 :

The dimension of ℘0

n K :

The dimension of ℘ K

\(\hat{n}_{K}\) :

The dimension of \(\hat{\mathcal{P}}_{K}\)

n j K :

The dimension of ℘ j K

\(\mathcal{N}\) :

The set of global nodes on V h

\(\hat{\mathcal{N}}\) :

The set of global nodes on \(\hat{V}_{h}\)

\(\mathcal{N}^{j}\) :

The set of global nodes on V j h

\(\mathcal{N}_{0}\) :

The set of local nodes on ℘0

\(\mathcal{N}_{K}\) :

The set of local nodes on ℘ K

\(\hat{\mathcal{N}}_{K}\) :

The set of local nodes on \(\hat{\mathcal{P}}_{K}\)

\(\mathcal{N}_{K}^{j}\) :

The set of local nodes on ℘ j K

ν 0 i :

A node on ℘0

ν K i :

A node on ℘ K

\(\hat{\nu}^{K}_{i}\) :

A node on \(\hat{\mathcal{P}}_{K}\)

ν K,j i :

A node on ℘ j K

℘0 :

The function space on K 0 for V h

\(\hat{\mathcal{P}}_{0}\) :

The function space on K 0 for \(\hat{V}_{h}\)

℘ j0 :

The function space on K 0 for V j h

℘ K :

The local function space on K for V h

\(\hat{\mathcal{P}}_{K}\) :

The local function space on K for \(\hat{V}_{h}\)

℘ j K :

The local function space on K for V j h

P q (K):

The space of polynomials of degree ≤q on K

\(\overline{\mathcal{P}}_{K}\) :

The local function space on K generated by {℘ j K } mj=1

R :

The residual, R(U)=A(U)−f

r :

The arity of the multilinear form a (the rank of A and A K)

U :

The discrete approximate solution, U≈u

(U i ):

The vector of expansion coefficients for U=∑ Ni=1 U i φ i

u :

The exact solution of the given model A(u)=f

V :

The space of trial functions on Ω (the trial space)

\(\hat{V}\) :

The space of test functions on Ω (the test space)

V h :

The space of discrete trial functions on Ω (the discrete trial space)

\(\hat{V}_{h}\) :

The space of discrete test functions on Ω (the discrete test space)

V j h :

A discrete function space on Ω

|V|:

The dimension of a vector space V

Φ i :

A basis function in ℘0

\(\hat{\Phi}_{i}\) :

A basis function in \(\hat{\mathcal{P}}_{0}\)

Φ j i :

A basis function in ℘ j0

φ i :

A basis function in V h

\(\hat{\phi}_{i}\) :

A basis function in \(\hat{V}_{h}\)

φ j i :

A basis function in V j h

φ K i :

A basis function in ℘ K

\(\hat{\phi}_{i}^{K}\) :

A basis function in \(\hat{\mathcal{P}}_{K}\)

φ K,j i :

A basis function in ℘ j K

φ :

The dual solution

\(\mathcal{T}\) :

The mesh

Ω:

A bounded domain in ℝd

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Logg, A. Automating the Finite Element Method. Arch Computat Methods Eng 14, 93–138 (2007). https://doi.org/10.1007/s11831-007-9003-9

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