On the Robustness of Two-Level Preconditioners for Quadratic FE Orthotropic Elliptic Problems
We study the construction of subspaces for quadratic FEM orthotropic elliptic problems with a focus on the robustness with respect to mesh and coefficient anisotropy. In the general setting of an arbitrary elliptic operator it is known that standard hierarchical basis (HB) techniques do not result in splittings in which the angle between the coarse space and its (hierarchical) complement is uniformly bounded with respect to the ratio of anisotropy. In this paper we present a robust splitting of the finite element space of continuous piecewise quadratic functions for the orthotropic problem. As a consequence of this result we obtain also a uniform condition number bound for a special sparse Schur complement approximation. Further we construct a uniform preconditioner for the pivot block with optimal order of computational complexity.
KeywordsAnisotropy Ratio Coarse Space Hierarchical Basis Spectral Condition Number Relative Condition Number
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