An MP-DWR method for $h$-adaptive finite element methods

TL;DR

This paper introduces a novel MP-DWR method for $h$-adaptive finite element methods that uses multiple-precision calculations to construct dual spaces, improving efficiency and storage over traditional $h$- and $p$-approaches.

Contribution

A new multiple-precision based approach for dual space construction in finite element methods, enhancing efficiency and storage compared to existing $h$- and $p$-approaches.

Findings

The MP-DWR approach is feasible with C++ templates.

It shows significant improvements in efficiency and storage.

Performance is comparable to higher order interpolation methods.

Abstract

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.