B-spline approximation of elliptic problems with non-smooth coefficients
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Andreas Keller [1]
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[1] University of Applied Sciences Würzburg
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- Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 10, Nº. 1-2, 2018, págs. 1-27
- Idioma: inglés
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Resumen
Weighted B-splines approximate smooth solutions of elliptic problems with maximal order. However, lack of regularity due to non-smooth coefficients of the partial differential equation can cause a severe loss of accuracy. A typical model problem is −∇ · (α∇u) = f in Ω, u = 0 on ∂Ω, where α is discontinuous across a submanifold Γ ⊂ Ω. We show for the twodimensional case, that the optimal convergence rates can be retained if we augment addition so-called singular splines to the weighted spline basis. The singular splines are constructed with an implicit representation of Γ and can model the discontinuous gradients of solutions accurately. As a result we obtain a meshless method of optimal order with the computational advantages of the B-spline calculus.
