Stability analysis for the penalty plus hybrid and the direct Trefftz methods for singularity problems

For solving the linear algebraic equations Ax = b, the new stability analysis is made based on the effective condition number Cond_eff. The Cond_eff may provide a better upper bound of relative errors of x resulting from the rounding errors of b, than the traditional condition number Cond, which with too large value is, in many times, misleading. In this paper, we apply the effective condition number to the Trefftz methods (TMs) for Poisson's equations with singularities. Two TMs, such as the penalty plus hybrid TM and the Lagrange multiple (i.e., direct) TM, are studied. We focus on the stability analysis of the solutions when the optimal superconvergence is achieved. When solving Motz's problem, the benchmark of singularity problems, by the penalty plus hybrid TM, we have derived that Cond _ eff = O (N (sqrt(2))^N) and Cond = O (N² 2^N), where N is the number of the singular particular functions used. For solving Motz's problem by the direct TM, we have derived that Cond / Cond _ eff = O (N sqrt(2)^N). Numerical experiments are provided to verify the stability analysis made. In summary, the two TMs are efficient, but the penalty plus hybrid TM is more recommended, due to simplicity of algorithms without extra-variables and less limitations in applications.