Meshless Galerkin methods using radial basis functions

Abstract

Abstract. We combine the theory of radial basis functions with the field of Galerkin methods to solve partial differential equations. After a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions.

An integrated-RBF technique based on Galerkin formulation for elliptic differential equations

“…Galerkin RBF techniques have been considered in [2,[12][13][14]. In those works, conventional RBF approximations were employed.…”

Solutions to Pseudodifferential Equations Using Spherical Radial Basis Functions

“…Since τ + α ≤ τ and τ + α ≤ s ≤ 2τ , Proposition 3.2 assures us that there exists v ∈ V φ X satisfying u 1 − v τ +α ≤ Ch ν 1 X u 1 s , where ν 1 = min{s − τ − α, −2α, 2τ + |s|}. It follows from Lemma 4.5 and the above inequality that u 1 − u T. D. Pham and T. Tran [12] Estimate (4.17) can be obtained by using the well-known duality argument. We include the proof here for completeness.…”

“…The use of spherical radial basis functions results in meshless methods which, in recent years, have grown in popularity [12,13]. These methods are an alternative to finite-element methods.…”

Solvability of partial differential equations by meshless kernel methods

“…Candidates for further analysis are the weak meshless local Petrov-Galerkin method (MLPG, [3,2,4]) or the generalized finite element method [7] based on techniques using partitions of unity [17,6]. Chances are good that such methods also work nicely for meshless kernel techniques, since they surely work for interpolation [25] and certain simple problems in weak form [24]. A new and promising development that incorporates multilevel techniques into meshless kernel methods introduces multiscale kernels [20].…”