Solving the inverse Stokes problems by the modified collocation Trefftz method and Laplacian decomposition

Abstract

In this paper, a boundary-type meshfree algorithm is proposed to accurately and stably deal with the two-dimensional inverse Stokes problems, which are highly ill-conditioned. Based on the Laplacian decomposition, the Stokes equations are recast as three Laplace equations. Then the modified collocation Trefftz method (MCTM), one of the most promising boundary-type meshless methods, is adopted to solve these three Laplace equations. The MCTM can stabilize the numerical scheme and obtain highly accurate results by utilizing the characteristic length. Accordingly, the numerical solutions of these three Laplace equations are expressed by linear combination of the modified T-complete functions. The unknown coefficients in the solution expressions are found by enforcing the satisfactions of the boundary conditions at the boundary collocation points. Three numerical examples are provided to show the efficacy and stability of the proposed meshless method. Besides, noises are added into the boundary conditions to demonstrate the stability of the proposed scheme for dealing with the inverse Stokes problems.