Balancing truncation and round-off errors in practical FEM: one-dimensional analysis

In finite element methods (FEMs), the accuracy of the solution cannot increase indefinitely because the round-off error increases when the number of degrees of freedom (DoFs) is large enough. This means that the accuracy that can be reached is limited. A priori information of the highest attainable accuracy is therefore of great interest. In this paper, we devise an innovative method to obtain the highest attainable accuracy. In this method, the truncation error is extrapolated when it converges at the analytical rate, for which only a few primary $h$-refinements are required, and the bound of the round-off error is provided through extensive numerical experiments. The highest attainable accuracy is obtained by minimizing the sum of these two types of errors. We validate this method using a one-dimensional Helmholtz equation in space. It shows that the highest attainable accuracy can be accurately predicted, and the CPU time required is much less compared with that using the successive $h$-refinement.