Abstract: "For a simple model problem -- the Laplace equation on the unit square with a Dirichlet boundary function vanishing for x = 0, x = 1, and y = 1, and equaling some suitable g(x) for y = 0 -- we present a proof of convergence for the combination technique, a modern, efficient, and easil...

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Format:	Buch
Sprache:	English
Veröffentlicht:	München 1993
Schriftenreihe:	Technische Universität <München>: TUM-I 9336
Schlagworte:	Differential equations, Partial
Zusammenfassung:	Abstract: "For a simple model problem -- the Laplace equation on the unit square with a Dirichlet boundary function vanishing for x = 0, x = 1, and y = 1, and equaling some suitable g(x) for y = 0 -- we present a proof of convergence for the combination technique, a modern, efficient, and easily parallelizable sparse grid solver for elliptic partial differential equations that recently gained importance in fields of applications like computational fluid dynamics. For full square grids with meshwidth h and O(h⁻²) grid points, the order O(h²) of the discretization error using finite differences was shown in [12], if g(x) [member of] C²[0,1]. In this paper, we show that the finite difference discretization error of the solution produced by the combination technique on a sparse grid with only O((h⁻¹ log₂ (h⁻¹)) grid points is of the order O(h² log₂ (h⁻¹)), if the Fourier coefficients b[subscript k] of g̃, the 2-periodic and 0-symmetric extension of g, fulfill [formula] for some arbitrary small positive [epsilon]. If 0 <[epsilon] [<or =] 1, this is valid for g [member of] C⁴[0,1] and g(0) = g(1) = g''(0) = g''(1) = 0, e.g.. A simple transformation even shows that g [member of] C⁴[0,1] is sufficient. Furthermore, we present results of numerical experiments with functions g of varying smoothness."
Beschreibung:	25 S. graph. Darst.