Adaptive meshfree solving of linear PDEs: Analysis of target-data dependent greedy kernel methods
Tizian Wenzel  1@  , Daniel Winkle  1  , Gabriele Santin  2  , Bernard Haasdonk  1  
1 : University of Stuttgart
2 : Bruno Kessler Foundation

We consider meshless solving of PDEs
Lu = f on Ω, u = g on ∂Ω (1)
via symmetric kernel collocation by using greedy kernel methods. In this way we avoid the need for a mesh
generation, which can be challenging for non-standard domains Ω or manifolds. We introduce and discuss
different kind of greedy selection criteria, such as the PDE-P -greedy and the PDE-f -greedy.
Subsequently we analyze the convergence rates of these algorithms and provide bounds on the approximation
error in terms of the number of greedily selected points. Especially we prove that target-data dependent
algorithms exhibit faster convergence rates.
The provided analysis is applicable to PDEs both on domains and manifolds. This and the advantages of
target-data dependent algorithms is highlighted by numerical examples.


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