Preconditioning and a posteriori error estimates using h‐ and p‐hierarchical finite elements with rectangular supports

We show some of the properties of the algebraic multilevel iterative methods when the hierarchical bases of finite elements (FEs) with rectangular supports are used for solving the elliptic boundary value problems. In particular, we study two types of hierarchies; the so‐called h‐ and p‐hierarchical FE spaces on a two‐dimensional domain. We compute uniform estimates of the strengthened Cauchy–Bunyakowski–Schwarz inequality constants for these spaces. Moreover, for diagonal blocks of the stiffness matrices corresponding to the fine spaces, the optimal preconditioning matrices can be found, which have tri‐ or five‐diagonal forms for h‐ or p‐refinements, respectively, after a certain reordering of the elements. As another use of the hierarchical bases, the a posteriori error estimates can be computed. We compare them in test examples for h‐ and p‐hierarchical FEs with rectangular supports. Copyright © 2008 John Wiley & Sons, Ltd.