TSFC: A Structure-Preserving Form Compiler

Abstract

A form compiler takes a high-level description of the weak form of partial differential equations and produces low-level code that carries out the finite element assembly. In this paper we present the Two-Stage Form Compiler (TSFC), a new form compiler with the main motivation to maintain the structure of the input expression as long as possible. This facilitates the application of optimizations at the highest possible level of abstraction. TSFC features a novel, structure-preserving method for separating the contributions of a form to the subblocks of the local tensor in discontinuous Galerkin problems. This enables us to preserve the tensor structure of expressions longer through the compilation process than other form compilers. This is also achieved in part by a two-stage approach that cleanly separates the lowering of finite element constructs to tensor algebra in the first stage, from the scheduling of those tensor operations in the second stage. TSFC also efficiently traverses complicated expressions, and experimental evaluation demonstrates good compile-time performance even for highly complex forms.Algorithm 1 Generated loop nests for the expression (4.2).1: for all q do 2:3: for all q, r do 4: 1) q,r * c r 5: for all q do 6:J (1,2) [q] ← 0 7: for all q, r do 8: 2) q,r * c r 9: for all q do 10: J (2,1) [q] ← 0 11: for all q, r do 12: J (2,1) [q] += C (2,1) q,r * c r 13: for all q do 14: J (2,2) [q] ← 0 15: for all q, r do 16: J (2,2) [q] += C (2,2) q,r * c r

Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere

“…The authors would like to thank John Thuburn for providing the semi-Lagrangian code that was used for the Williamson 5 convergence test, and the Firedrake project [Rathgeber et al, 2016, Homolya et al, 2017, Kirby and Mitchell, 2017, Gibson et al, 2018, along with PETSc [Balay et al, 2016[Balay et al, , 1997] and other upstream tools [Hendrickson and Leland, 1995, Dalcin et al, 2011] for making the code development in this paper possible. This research was conducted with support from Natural Environment Research Council grants NE/I000747/1 and NE/K006789/1.…”

A general approach to transforming finite elements

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An Augmented Lagrangian Preconditioner for the 3D Stationary Incompressible Navier--Stokes Equations at High Reynolds Number

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