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In finite element methods (FEMs), the accuracy of the solution cannotincrease indefinitely because the round-off error increases when the number ofdegrees of freedom (DoFs) is large enough. This means that the accuracy thatcan be reached is limited. A priori information of the highest attainableaccuracy is therefore of great interest. In this paper, we devise an innovativemethod to obtain the highest attainable accuracy. In this method, thetruncation error is extrapolated when it converges at the analytical rate, forwhich only a few primary $h$-refinements are required, and the bound of theround-off error is provided through extensive numerical experiments. Thehighest attainable accuracy is obtained by minimizing the sum of these twotypes of errors. We validate this method using a one-dimensional Helmholtzequation in space. It shows that the highest attainable accuracy can beaccurately predicted, and the CPU time required is much less compared with thatusing the successive $h$-refinement.

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