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Title: Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm

Journal Article · · SIAM Journal on Matrix Analysis and Applications
DOI:https://doi.org/10.1137/18m1165190· OSTI ID:1669216
 [1];  [2]
  1. Tufts Univ., Medford, MA (United States). Dept. of Mathematics
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing; Portland State Univ., OR (United States). Dept. of Mathematics
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Algebraic multigrid (AMG) coarse spaces are commonly constructed so that they exhibit the so-called weak approximation property (WAP) which is a necessary and sufficient condition for uniform two-grid convergence. Here, this paper studies a modification of such coarse spaces so that the modified ones provide approximation in energy norm. Our modification is based on the projection in energy norm onto an orthogonal complement of original coarse space. This generally leads to dense modified coarse space matrices, which is hence computationally infeasible. To remedy this, based on the fact that the projection involves inverse of a well-conditioned matrix, we use polynomials to approximate the projection and, therefore, obtain a practical, sparse modified coarse matrix and prove that the modified coarse space maintains computationally feasible approximation in energy norm. We present some numerical results for both PDE discretization matrices as well as graph Laplacian ones, which are in accordance with our theoretical results.

Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
Grant/Contract Number:
AC52-07NA27344; DMS-1619640; DMS-1620063
OSTI ID:
1669216
Report Number(s):
LLNL-JRNL-744422; 899470
Journal Information:
SIAM Journal on Matrix Analysis and Applications, Vol. 40, Issue 3; ISSN 0895-4798
Publisher:
SIAMCopyright Statement
Country of Publication:
United States
Language:
English

References (25)

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Related Subjects

97 MATHEMATICS AND COMPUTING
AMG
strong approximation property
weak approximation property