A Meshfree Method for Solving the Stokes Problem on Bounded Domains

Abstract

We investigate the application of an analytically divergence-free kernel method for solving the Stokes equations on bounded domains in this paper. We develop a new collocation method based on the divergence-free kernel trial spaces produced by radial basis functions (RBFs). An

L^2

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error is obtained when the testing discretization is finer than the trial discretization. The convergence rates depend on the regularity of the solutions, the smoothness of the computational domain, the selection of the scaling parameter, and the approximation of divergence-free kernel trial spaces (for the velocity field) and RBF trial spaces (for the pressure field). The convergence theory is established in Sobolev spaces, which covers stationary and non-stationary approximations. Several numerical examples are provided to demonstrate numerical efficiency.

Keywords

Stokes problem / Radial basis functions (RBFs) / Collocation method / Divergence-free kernel / Sobolev spaces / Convergence rates / 65N12 / 65N15 / 65N35

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