  In a dual weighted residual method based on the finite element framework, the
Galerkin orthogonality is an issue that prevents solving the dual equation in
the same space as the one for the primal equation. In the literature, there
have been two popular approaches to constructing a new space for the dual
problem, i.e., refining mesh grids ($h$-approach) and raising the order of
approximate polynomials ($p$-approach). In this paper, a novel approach is
proposed for the purpose based on the multiple-precision technique, i.e., the
construction of the new finite element space is based on the same configuration
as the one for the primal equation, except for the precision in calculations.
The feasibility of such a new approach is discussed in detail in the paper. In
numerical experiments, the proposed approach can be realized conveniently with
C++ \textit{template}. Moreover, the new approach shows remarkable improvements
in both efficiency and storage compared with the $h$-approach and the
$p$-approach. It is worth mentioning that the performance of our approach is
comparable with the one through a higher order interpolation ($i$-approach) in
the literature. The combination of these two approaches is believed to further
enhance the efficiency of the dual weighted residual method.
