Automating the Finite Element Method

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. Abbreviations A The differential operator of the model A(u)= f A The global tensor with entries {Ai}i∈I A0 The reference tensor with entries {Aiα}i∈IK,α∈A Ā0 The matrix representation of the (flattened) reference tensor A0 A The element tensor with entries {Ai }i∈IK a The semilinear, multilinear or bilinear form aK The local contribution to a multilinear form a from K A. Logg ( ) Scientific Computing, Simula Research Laboratory, PO Box 134, 1325 Lysaker, Norway e-mail: [email protected] a The vector representation of the (flattened) element tensor A A The set of secondary indices B The set of auxiliary indices e The error, e=U − u FK The mapping from K0 to K GK The geometry tensor with entries {GK }α∈A gK The vector representation of the (flattened) geometry tensor GK I The set ∏r j=1[1,N ] of indices for the global tensor A IK The set ∏r j=1[1, njK ] of indices for the element tensor A (primary indices) ιK The local-to-global mapping from NK to N ι̂K The local-to-global mapping from N̂K to N̂ ι j K The local-to-global mapping from N j K to N j K A cell in the mesh T K0 The reference cell L The linear form (functional) on V̂ or V̂h m The number of discrete function spaces used in the definition of a N The dimension of V̂h and Vh N The dimension V j h Nq The number of quadrature points on a cell n0 The dimension of P0 nK The dimension of PK n̂K The dimension of P̂K n j K The dimension of P j K N The set of global nodes on Vh N̂ The set of global nodes on V̂h N j The set of global nodes on V j h N0 The set of local nodes on P0 NK The set of local nodes on PK N̂K The set of local nodes on P̂K