Approximate solutions of partial differential equations by some Meshfree Greedy Algorithms

Abstract

In this article, we use some greedy algorithms to avoid the ill‐conditioning of the final linear system in unsymmetric Kansa collocation method. The greedy schemes have the same background, but we use them in different settings. In the first algorithm, the optimal trial points for interpolation obtained among a huge set of initial points are used for numerical solution of partial differential equations (PDEs). In the second algorithm, based on the Kansa's method, the PDE is discretized to a finite number of test functional equations, and a greedy sparse discretization is applied for approximating the linear functionals. Each functional is stably approximated by some few trial points with an acceptable accuracy. The third greedy algorithm is used to generate the test points. This paper shows that the greedily selection of nodes yields a better conditioning in contrast with usual full meshless methods. Some well‐known PDE examples are solved and compared with the full unsymmetric Kansa's technique. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1884–1899, 2017

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