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

Abstract

In this paper, the harmonic functions of Laplace's equations are derived explicitly for the Dirichlet and the Neumann boundary conditions on the boundary of a sector. Those harmonic functions are more explicit than those of Volkov [Volkov EA, Block method for solving the Laplace equation and for constructing conformal mappings. Boca Raton: CRC Press; 1994], and easier to expose the mild singularity at the domain corners of the Laplace solutions. Moreover, the particular solutions of Poisson's equation on the polygon is also provided. We also explore in detail the singularities of the polygons with the boundary angles Θ=π/2, 3π/2, π and 2π, which often occur in many testing models. 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.