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Reduced order modelling, in which a full system response is projected onto a subspace of lower dimensionality, has been used previously to accelerate finite element solution schemes by reducing the size of the involved linear systems. In the present work we take advantage of a secondary effect of such reduction for explicit analyses, namely that the stable integration time step is increased far beyond that of the full system. This phenomenon alleviates one of the principal drawbacks of explicit methods, compared with implicit schemes. We present an explicit finite element scheme in which time integration is performed in a reduced basis. Futhermore, we present a simple procedure for imposing inhomogeneous essential boundary conditions, thus overcoming one of the principal deficiencies of such approaches. The computational benefits of the procedure within a GPU-based execution framework are examined, and an assessment of the errors introduced is given. It is shown that speedups approaching an order of magnitude are feasible, without introduction of prohibitive errors, and without hardware modifications. The procedure may have applications in interactive simulation and medical image-guidance problems, in which both speed and accuracy are vital.