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  1. 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 , H (curl), or H (div)-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. 
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  2. 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. 
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  3. null (Ed.)