Solving inverse Stokes problems by modified collocation Trefftz method

Abstract

In this paper, the two-dimensional inverse Stokes problems, governed by bi-harmonic equations, are stably solved by the modified collocation Trefftz method (MCTM). In some practical applications of the Stokes problems, part of the boundary conditions cannot be measured in advance, so the mathematical descriptions of such problems are known as the inverse Stokes problems. When numerical simulation is adopted for solutions of the inverse Stokes problems, the solutions will become extremely unstable, which means that small perturbations in the boundary conditions will result in large errors of the final results. Hence, we adopted the MCTM for stably and efficiently analyzing the inverse Stokes problems. The MCTM is one kind of boundary-type meshless methods, so the mesh generation and the numerical quadrature can be avoided. Besides, the numerical solution is expressed as a linear combination of T-complete functions modified by a characteristic length. By enforcing the satisfactions of the boundary conditions at every boundary node, a system of linear algebraic equations will be yielded. The unknown coefficients in the solution expression can be acquired by directly inverting the coefficient matrix. The numerical solutions and their derivatives can be easily obtained by linear summation. Three numerical examples are provided to demonstrate the accuracy and the stability of the proposed meshless scheme for solving the two-dimensional inverse Stokes problems.