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Terascale implicit methods for partial differential equations. (English) Zbl 1018.65121

Feng, Xiaobing (ed.) et al., Recent advances in numerical methods for partial differential equations and applications. Proceedings of the 2001 John H. Barrett memorial lectures trends in computational mathematics, Univ. of Tennessee, Knoxville, TN, USA, May 10-12, 2001. Providence, RI: AMS, American Mathematical Society. Contemp. Math. 306, 29-84 (2002).
From the introduction: Section 2 describes the algorithmic core, Newton-Krylov-Schwarz (NKS), as well as Schur preconditioning, a domain-decomposed alternative to Schwarz. Various extensions to this core, related to the large-scale numerical analysis of nonlinear partial differential equations (PDEs) are described in Section 3. Section 4 makes a brief foray into the field of optimization subject to large-scale PDE-based constraints. This is built upon Schur preconditioning applied algebraically, with Schwarz applied inside of Schur in the usual subdomain-by-subdomain sense.
Section 5 briefly highlights a Gordon Bell prize-winning computation based on NKS algorithmics. This has been thoroughly documented elsewhere especially in its aspect of high performance, but is summarized here to emphasize the practical rewards of an NKS approach. Finally, in Section 6 we describe the major goals of a five-year, nine-institution project, “Terascale Optimal PDE Simulations,” which is one of the seven “Integrated Software Infrastructure Centers” of the DOE’s “Scientific Discovery through Advanced Computing” (SciDAC) initiative, launched in 2001. This project is built on top of a significant base of existing software, including the PETSc library from Argonne National Laboratory and the Hypre library from Lawrence Livermore National Laboratory.
For the entire collection see [Zbl 1001.00031].

MSC:

65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
65H10 Numerical computation of solutions to systems of equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q30 Navier-Stokes equations
76M10 Finite element methods applied to problems in fluid mechanics
65Y05 Parallel numerical computation
76D05 Navier-Stokes equations for incompressible viscous fluids
76N15 Gas dynamics (general theory)
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