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A generic grid interface for parallel and adaptive scientific computing. I: Abstract framework. (English) Zbl 1151.65089

Summary: We give a mathematically rigorous definition of a grid for algorithms solving partial differential equations. Unlike previous approaches [cf. W. Benger, Visualization of general relativistic tensor fields via a fiber bundle data model (2004; Zbl 1096.83508); G. Berti, Generic software components for scientific computing (2000; Zbl 0972.68506)], our grids have a hierarchical structure. This makes them suitable for geometric multigrid algorithms and hierarchical local grid refinement. The description is also general enough to include geometrically non-conforming grids. The definitions in this article serve as the basis for an implementation of an abstract grid interface as C++ classes in the framework of part II of this paper [ibid. 82, No. 2–3, 121–138 (2008; Zbl 1151.65088)].

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65Y05 Parallel numerical computation

Software:

GrAL; DUNE
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Full Text: DOI

References:

[1] Bastian P, Blatt M, Dedner A, Engwer C, Klöfkorn R, Kornhuber R, Ohlberger M, Sander O (2008) A generic grid interface for parallel and adaptive scientific computing. Part II: Implementation and tests in DUNE. Computing (this issue) · Zbl 1151.65088
[2] Benger W (2005) Visualization of general relativistic tensor fields via a Fiber Bundle Data Model. PhD thesis, Freie Universität Berlin · Zbl 1096.83508
[3] Berti G (2000) Generic software components for scientific computing. PhD thesis, BTU Cottbus · Zbl 0972.68506
[4] Botta N, Ionescu C, Linstead C, Klein R (2006) Structuring distributed relation-based computations with SCDRC. Technical report, PIK Report No. 103, Potsdam Institute for Climate Impact Research
[5] DUNE – Distributed and Unified Numerics Environment. http://dune-project.org/
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