Local Dirichlet problems on subdomains of decomposition in $HP$-discretizations, and optimal algorithms for their solution

We study the key component for the DD (domain decomposition) algorithms for $hp$ discretizations of second order elliptic partial differential equations. It is related to solving local discrete Dirichlet problems on subdomains of decomposition and is the most time consuming part of DD algorithm. We suggest an optimal (except for the operation of multiplication by an element stiffness matrix unavoidable in any iterative process) solver for these local Dirichlet problems. When incorporated in our earlier developed algorithms, this solver directly leads to the optimal DD solver for the global system of finite element algebraic equations.