Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA, USA

Johannes Kraus

Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA, USA

and

University for Mining and Metallurgy, Leoben, Austria

Panayot S. Vassilevski

Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA, USA

Abstract

Recently a new general algorithm, denoted "element-free AMGe," was proposed [1], for constructing the interpolation weights in algebraic multigrid. The method uses an extension mapping to provide boundary values outside a neighborhood about a fine-grid degree of freedom (dof) to which interpolation is desired. The interpolated value is obtained by a matrix-dependent harmonic extension of these boundary values into the interior of the neighborhood. In essence, this method is designed to capture information that can be obtained from individual finite-element stiffness matrices, as is done in the so-called "element-based AMG method" (AMGe) [2], for problems in which such matrices may not be available.

The object of such a method is to characterize, on a local scale, the nature of the smooth modes of the global operator, insuring that they can be represented by the interpolation. We propose here a modification to this method, in which we use a more global method of localizing the nature of the smooth modes. In essence, the "extended neighborhood" of the fine-grid dof is defined globally, by looking at the coarse dofs on every level that feed information to the specified fine dof. The harmonic extension is then generated from a "multilevel" extended neighborhood, which can be constructed to allow for recursive improvement of the interpolation operators as each coarser level is addressed.

We describe the basic algorithm and methods of implementation, and discuss some early experimental results.

This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.

[1] Henson, V. E., and Vassilevski, P. S., "Element Free AMGe: general
algorithms for computing interpolation weights in AMG," to appear in
*SIAM Journal on Scientific Computing*.

[2] Brezina, M., Cleary, A. J., Falgout, R. D., Henson, V. E., Jones, J.
E., Manteuffel, T. A., McCormick, S. F., and Ruge, J. W., "Algebraic
multigrid based on element interpolation (AMGe)," *SIAM Journal on
Scientific Computing*, V. 22, pp. 1570-1592, 2000.