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

In finite element methods, the accuracy of the solution cannot increase indefinitely since the round-off error related to limited computer precision increases when the number of degrees of freedom (DoFs) is large enough. Because a priori information of the highest attainable accuracy is of great interest, we construct an innovative method to obtain the highest attainable accuracy given the order of the elements. In this method, the truncation error is extrapolated when it converges at the asymptotic rate, and the bound of the round-off error follows from a generically valid error estimate, obtained and validated 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 smaller compared with that using successive grid refinement.