Abstract: In this article, two kinds of expandable parallel finite element methods, based on two-grid discretizations, are given to solve the linear elliptic problems. Compared with the classical local and parallel finite element methods, there are two attractive features of the methods shown in this article: 1) a partition of unity is used to generate a series of local and independent subproblems to guarantee the final approximation globally continuous; 2) the computational domain of each local subproblem is contained in a ball with radius of $O(H)$ ($H$ is the coarse mesh parameter), which means methods in this article are more suitable for parallel computing in a large parallel computer system. Some a priori error estimation are obtained and optimal error bounds in both $H^1$-normal and $L^2$-normal are derived. Finally, numerical results are reported to test and verify the feasibility and validity of our methods.

Key words: Two-grid method, expandable method, partition of unity, parallel algorithm, finite element method