Application of the AmgX Library to the Discontinuous Galerkin Methods for Elliptic Problems

• Published in: Parallel Computational Technologies Vol. 1437; pp. 178 - 193
• Main Authors: Evstigneev, N. M., Ryabkov, O. I.
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2021; Springer International Publishing
• Series: Communications in Computer and Information Science
• Subjects: AmgX library; Darcy problem; Discontinuous Galerkin methods; Elliptic problems; Multigrid methods
• ISBN: 9783030816902; 3030816907
• ISSN: 1865-0929; 1865-0937
• DOI: 10.1007/978-3-030-81691-9_13

We consider an application of the AmgX library by NVIDIA as the preconditioner or solver for discrete elliptic problems expressed through Discontinuous Galerkin methods (DG) with various formulations. The effect of poor geometric multigrid performance on the elliptic DG formulation has been discussed in a recent paper by Fortunato, Rycroft, and Saye. In the present study, we check the ‘out-of-the-box’ performance of the Algebraic Multigrid Method (AMG) implemented in the open-source variant of the AmgX library. Four different DG discretization schemes are considered, namely local DG, compact DG, Bassi–Rebay-2 scheme, and internal penalty methods, including symmetric and nonsymmetric formulations. The local DG scheme is considered in its dual form; the rest are considered in primal form. All these methods yield a block matrix with a compact stencil, which is passed to the AmgX library (or Krylov-subspace methods with the AmgX library used as a preconditioner) for the solution of the linear system. We show that the library requires some code adjustments and additions before we can apply it to the block matrices by hand. It is also shown that the convergence of the AMG and Krylov-AMG methods is relatively poor and requires a reformulation of the problem. Further research is expected.