The meshless solutions of Laplacian non-harmonic and Cauchy problems by developing novel hybrid methods

Abstract

In the first part of the paper a non-harmonic boundary value problem of the Laplace equation is solved by new hybrid methods. We propose four hybrid methods, namely the Trefftz-Trefftz method (TTM), the Trefftz-MFS method (TMM), the MFS-Trefftz method (MTM), and the MFS-MFS method (MMM), by supplementing extra linear equations with the Trefftz functions or the fundamental solutions as the testers in Green’s second identity. In doing so the range of the coefficient matrix can be enlarged such as more close to the input vector, and meanwhile the accuracy is improved. In the second part we improve the accuracy of the MFS, which deteriorates due to the ill-posed property of the Cauchy problem. We derive two simple merit functions and minimize them to determine the source points in the MFS, which can further enhance the accuracy obtained by two optimal MTMs. These new hybrid methods apparently improve the accuracy than the MFS and the Trefftz method, when we solve the non-harmonic and Cauchy problems of the Laplace equation.