An effectively modified direct Trefftz method for 2D potential problems considering the domain's characteristic length

The present paper proposes a new modification of the direct Trefftz method by taking the characteristic length of the problem domain into account, whose inclusion into the T-complete bases ensures that the modified direct Trefftz method is stable, because the condition number of the resulting linear equations system can be greatly reduced over 12 orders. Then, the boundary element method and the Fourier series method are used to derive the linear equations system to determine the unknown coefficients, which can be employed to solve the mixed-boundary value 2D potential problems. We use numerical examples to explore why the conventional direct Trefftz method is unstable and the modified one is stable and workable. The direct Trefftz method is applicable to the case where the problem size is smaller or with its maximum length near to 1 and using suitable elements number or bases number. Under this condition, the modified method still has a great advantage to improve the accuracy up to two or three orders.