Gibbs Phenomena for L^q-Best Approximation in Finite Element Spaces -- Some Examples
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Recent developments in the context of minimum residual finite element methods are paving the way for designing finite element methods in non-standard function spaces. This, in particular, permits the selection of a solution space in which the best approximation of the solution has desirable properties. One of the biggest challenges in designing finite element methods are non-physical oscillations near thin layers and jump discontinuities. In this article we investigate Gibbs phenomena in the context of L^q-best approximation of discontinuities in finite element spaces with 1≤ q<∞. Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as q tends to 1. The aim here is to show the potential of L^1 as a solution space in connection with suitably designed meshes.
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