  We consider a sketched implementation of the finite element method for
elliptic partial differential equations on high-dimensional models. Motivated
by applications in real-time simulation and prediction we propose an algorithm
that involves projecting the finite element solution onto a low-dimensional
subspace and sketching the reduced equations using randomised sampling. We show
that a sampling distribution based on the leverage scores of a tall matrix
associated with the discrete Laplacian operator, can achieve nearly optimal
performance and a significant speedup. We derive an expression of the
complexity of the algorithm in terms of the number of samples that are
necessary to meet an error tolerance specification with high probability, and
an upper bound for the distance between the sketched and the high-dimensional
solutions. Our analysis shows that the projection not only reduces the
dimension of the problem but also regularises the reduced system against
sketching error. Our numerical simulations suggest speed improvements of two
orders of magnitude in exchange for a small loss in the accuracy of the
prediction.
