The partition of unity parallel finite element algorithm

This paper presents a partition of unity parallel finite element algorithm. This algorithm localizes the global residual problem of two grid method into some parallel local sub-problems, and use a simple partition of unity to assemble all the local solutions together. An oversampling technique is used and analyzed to decrease the undesirable effect of the artificial homogeneous Dirichlet boudary condition of local sub-problems. The analysis shows the error of this algorithm decays exponentially with respect to the oversampling parameter. Specially, on a regular coarse triangulation τ_H with mesh size H, an oversampling of diameter Hlog(1/H)$H \log (1/H)$ is sufficient to preserve the optimal convergence order. Numerical results verify the theoretical analysis.