A preconditioner for the $h$-$p$ version of the finite element method in two dimensions

Summary.

A preconditioner, based on a two-level mesh and a two-level orthogonalization, is proposed for the $h$-$p$ version of the finite element method for two dimensional elliptic problems in polygonal domains. Its implementation is in parallel on the subdomain level for the linear or bilinear (nodal) modes, and in parallel on the element level for the high order (side and internal) modes. The condition number of the preconditioned linear system is of order $\max\limits_i(1+\ln {{\textstyle H_ip_i}\over{\textstyle h_i}})^2$, where $H_i$ is the diameter of the $i$-th subdomain, $h_i$ and $p_i$ are the diameter of elements and the maximum polynomial degree used in the subdomain. This result reduces to well-known results for the $h$-version (i.e. $p_i=1$) and the $p$-version (i.e. $h_i=H_i$) as the special cases of the $h$-$p$ version.