In this talk we introduce new finite element spaces that can be constructed for agglomerates of standard elements in 3--d. The agglomerates are assumed to have certain regular structure in the sense that they share faces with closed boundaries composed of 1-d edges. The spaces are subspaces of a originally given de Rham sequence of respective $H^1$-conforming, $H(curl)$--conforming, $H(div)$--conforming and piecewise constant spaces. The procedure can be recursively applied so that a sequence of nested de Rham complexes can be constructed. As an application, we use the sequence of nested counterparts of the respective piecewise linears, the lowest order Nedelec, and the lowest order Raviart--Thomas spaces, to construct V--cycle multigrid as preconditioners in the conjugate gradient method. The resulting element agglomeration AMG (algebraic multigrid) methods appear to perform very similarly to the geometric MG in the case of uniformly refined meshes.

This talk is based on a joint work with Joseph E. Pasciak, Texas A&M University.

The work of this author was performed under the auspices of the U. S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract W-7405-Eng-48.