Particular solutions of Laplace's equations on polygons and new models involving mild singularities

Besides, the popular singularity models, Motz's and the cracked beam problems in Lu et al. [Lu TT, Hu HY, Li ZC. Highly accurate solutions of Motz's and the cracked beam problems. Eng Anal Bound Elem; 2004, in press], we design two new singularity models, one with discontinuous singularity, and the other with crack plus mild singularities. The collocation Trefftz method, the Schwarz alternating method, and their combinations may be chosen to seek the solution with high accuracy, which may be used to test other numerical methods. The particular solutions of the Laplace equations and their singularities are fundamental to numerical partial differential equations in both algorithms and error analysis.