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

In a dual weighted residual based adaptive finite element method for solving partial differential equations, a new finite element space needs to be built for solving the dual problem, due to the Galerkin orthogonality. Two popular approaches in the literature are $h$-refinement and $p$-refinement for the purpose, respectively, which would cause nonignorable requirements on computational resources. In this paper, a novel approach is proposed for implementing the dual weighted residual method through a multiple precision technique, i.e., the primal and dual problems are solved in two finite element spaces, respectively, built with the same mesh, degrees of freedom, and basis functions, yet different precisions. The feasibility of such an approach is discussed in detail. Besides effectively avoiding the issue introduced by the Galerkin orthogonality, remarkable benefits can also be expected for the efficiency from this new approach, since i). operations on reorganizing mesh grids and/or degrees of freedom in building a finite element space in $h$- and $p$-refinement methods can be avoided, and ii). both CPU time and storage needed for solving the derived system of linear equations can be significantly reduced compared with $h$- and $p$-refinement methods. In coding the algorithm with a library AFEPack, no extra module is needed compared with the program for solving the primal problem with the employment of a C++ feature $template$, based on which the floating point precision is used as a parameter of the template. A number of numerical experiments show the effectiveness and improvement of the efficiency of our method. Further discussion towards the limitation of the method is also delivered.