document: The test cases confirm that the high-order discretisation retains exponential convergence properties with increasing geometric and expansion polynomial order if both the solution and true surface are smooth. Errors are found to saturate when the geometric errors, due to the parametrisation of the surface elements, begin to dominate the temporal and spatial discretisation errors. For the smooth solutions considered as test cases, the results show that this dominance of geometric errors quickly limits the effectiveness of further increases in the number of degrees of freedom, either through mesh refinement or higher solution polynomial orders. Increasing the order of the geometry parametrisation reduces the geometric error. The analytic test cases presented here use a coarse curvilinear mesh; for applications, meshes are typically more refined in order to capture features in the solution and so will better capture the geometry and consequently reduce this lower bound on the solution error. If the solution is not smooth, we do not expect to see rapid convergence. In the case that the solution is smooth, but the true surface is not, then exponential convergence with P and Pg can only be achieved if, and only if, the discontinuities are aligned with element boundaries. However, if discontinuities lie within an element, convergence will be limited by the geometric approximation, since the true surface cannot be captured. In the cardiac problem, we consider both the true surface and solution to be smooth.
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