Domain decomposition preconditioning in the hierarchical \(p\)-version of the finite element method
The \(p\)-version of the finite element method for solving linear second-order elliptic equations in an arbitrary sufficiently smooth domain is studied in the framework of the domain decomposition (DD) method. Curvilinear elements associated with the square reference elements and satisfying conditions of generalized quasiuniformity are used to approximate the boundary and the boundary conditions. Two types of square reference elements are primarily considered with the products of integrated Legendre polynomials for coordinate functions. Condition number estimates are given, and preconditioning of the problems arising on subdomains and of the Schur complement together with the derivation of the global DD preconditioner are all considered. The authors obtain several DD preconditioners for which the generalized condition numbers vary from \(O((\log p)^3)\) to \(O(1)\).
scientific article; zbMATH DE number 1311308
scientific article