Wavelet-Like Methods in the Design of Efficient Multilevel Preconditioners for Elliptic PDEs

Abstract This article introduces a stabilization of the classical hierarchical basis (HB) method by modifying the HB functions using some computationally feasible approximate L 2 -projections onto finite element spaces of coarser levels. It is proved that the corresponding multilevel additive and multiplicative algorithms give spectrally equivalent preconditioners for model elliptic problems. In practical computation, one action of the proposed preconditioner is of optimal order algebraically. Numerical results are included to illustrate the efficiency and stability of the new preconditioning methods.

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