Robust Balanced Semi-coarsening AMLI Preconditioning of Biquadratic FEM Systems.

In the present study we demonstrate the construction of a robust multilevel preconditioner for biquadratic FE elliptic problems. In the general setting of an arbitrary elliptic operator it is well known that the standard hierarchical basis two-level splittings for higher order FEM elliptic systems deteriorate with increasing the anisotropy ratio. An alternative approach resulting in a robust hierarchical two-level splitting of the finite element space of continuos piecewise biquadratic functions involves the semi-coarsening mesh procedure. This evokes us to analyze the behavior of the constant in the strengthened CBS inequality, which is a quality measure for hierarchical two-level splittings of the FEM stiffness matrices, for the particular case of balanced semi-coarsening mesh refinement. We present new theoretical estimates which further we support by numerically computed CBS constants over a rich set of parameters (coarsening factor and anisotropy ratio). An optimal order multilevel algorithm is constructed on the basis of the proven uniform estimates and the theory of the Algebraic MultiLevel Iteration (AMLI) methods. Its total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio [ABSTRACT FROM AUTHOR]