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

Abstract. One approach to achieving correct finite element assembly is to ensure that the local orientation of facets relative to each cell in the mesh is consistent with the global orientation of that facet. Rognes et al. have shown how to achieve this for any mesh composed of simplex elements, and deal.II contains a serial algorithm for constructing a consistent orientation of any quadrilateral mesh of an orientable manifold. The core contribution of this paper is the extension of this algorithm for distributed memory parallel computers, which facilitates its seamless application as part of a parallel simulation system. Furthermore, our analysis establishes a link between the well-known Union-Find algorithm and the construction of a consistent orientation of a quadrilateral mesh. As a result, existing work on the parallelization of the Union-Find algorithm can be easily adapted to construct further parallel algorithms for mesh orientations.

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