Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 39, Issue 6, Pages A2538-A2563Publisher
SIAM PUBLICATIONS
DOI: 10.1137/17M1118087
Keywords
radial basis function; partition of unity; least squares; partial differential equation; Poisson equation; RBF-PUM
Categories
Funding
- European Commission CORDIS Marie Curie FP7 program grant [235730]
- National Science Foundation DMS grant [1552238]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1552238] Funding Source: National Science Foundation
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Recently, collocation-based radial basis function (RBF) partition of unity methods (PUMs) for solving partial differential equations have been formulated and investigated numerically and theoretically. When combined with stable evaluation methods such as the RBF-QR method, high order convergence rates can be achieved and sustained under refinement. However, some numerical issues remain. The method is sensitive to the node layout, and condition numbers increase with the refinement level. Here, we propose a modified formulation based on least squares approximation. We show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling. We derive theoretical error estimates both for the collocation and least squares RBF-PUMs. Numerical experiments are performed for the Poisson equation in two and three space dimensions for regular and irregular geometries. The convergence experiments confirm the theoretical estimates, and the least squares formulation is shown to be 5-10 times faster than the collocation formulation for the same accuracy.
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