Applications of the modified Trefftz method for the Laplace equation

In this paper, the Laplace problems are solved by using the modified Trefftz method. For discontinuous boundary problem, singular problem, and degenerate scale problem, the conventional Trefftz method encounters the numerical instability owing to rank deficiency and high-order T-complete functions. To overcome these problems, the characteristic length of problem domain and high-order T-complete functions is introduced to obtain a modified Trefftz method, equipping with a characteristic length factor to make sure that this method is stable. Besides, the high-order T-complete functions can be clearly described for the discontinuous boundary conditions. Comparing with the solutions in the previous literature, the present method is powerful even for the problem with complex boundary and with adding random noise on the boundary data. It is also successfully solving the degenerate scale problem, resulting to a highly accurate result never seen before.