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The numerical solution of partial differential equations is at the heart ofmany grand challenges in supercomputing. Solvers based on high-orderdiscontinuous Galerkin (DG) discretisation have been shown to scale on largesupercomputers with excellent performance and efficiency, if the implementationexploits all levels of parallelism and is tailored to the specificarchitecture. However, every year new supercomputers emerge and the list ofhardware-specific considerations grows, simultaneously with the list of desiredfeatures in a DG code. Thus we believe that a sustainable DG code needs anabstraction layer to implement the numerical scheme in a suitable language. Weexplore the possibility to abstract the numerical scheme as small tensoroperations, describe them in a domain-specific language (DSL) resembling theEinstein notation, and to map them to existing code generators which generatesmall matrix matrix multiplication routines. The compiler for our DSLimplements classic optimisations that are used for large tensor contractions,and we present novel optimisation techniques such as equivalent sparsitypatterns and optimal index permutations for temporary tensors. Our applicationexamples, which include the earthquake simulation software SeisSol, show thatthe generated kernels achieve over 50 % peak performance while the DSLconsiderably simplifies the implementation.

Yet Another Tensor Toolbox for discontinuous Galerkin methods and other applications