Domain decomposition preconditioning in the hierarchical p -version of the finite element method

Abstract The p-version 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 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 derivation of the global DD preconditioner are all considered. We obtain several DD preconditioners for which the generalized condition numbers vary from O((log p)3) to O(1). The paper consists of seven sections. We give some preliminary results for the 1D case, condition number estimates and some inequalities for the 2D reference element. The preconditioning of the Schur complement is detailed for the 2D reference element, the p-version with curvilinear finite elements is considered next, and the DD preconditioning of the entire stiffness matrix is introduced and analyzed. Some related reference elements using Lobatto-Chebyshev nodal bases on the boundary are introduced and studied with respect to the effect on the preconditioner—the (log p)3 factor may thus be removed. We discuss also some specific features of the algorithms stemming from the suggested preconditioners, which provide a low computational cost and a high degree of parallelization.

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