科研之友
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摘要
The Laplace equation can be transformed to the first kind boundary integral equations (BIEs), and the corner and discontinuity singularity of Laplace's equation can be studied by the open arcs of the BIEs. For the first kind BIEs, there exist the Galerkin method (GM) and the collocation method (CM): but they suffer in low convergence. The advanced (i.e., the mechanical) quadrature methods (AQMs) and the splitting extrapolation methods (SEMs) originated in [16,17] are proposed in Huang et al. [15] for first kind BIEs with open arcs, to achieve O(h(3)) or even O(h(5)) convergence, and the excellent stability with Cond. = O(h(-1)), where h is the uniform meshspacing, accompanied with the strict analysis. Moreover, the algorithms of AQMs and the SEMs are simple without any integration computation. Hence the AQMs and the SEMs are superior to the existing methods, such as GM and CM. A challenging discontinuity model of Laplace's equation is proposed in Li et al. [20], and the collocation Trefftz method (CTM) is used to give highly accurate solutions. For the AQMs and the SEMs, the strict theoretical analysis is given in [15], and the discontinuity model [20] is dealt with in this paper, to also achieve highly accurate solutions. The numerical solutions in this paper display that the AQMs and the SEMs are significant not only to the first kind BIEs with the open arc singularity, but also to Laplace's equation with highly strong singularity such as the discontinuity of solutions. Moreover, the link of the first kind BIEs with open arcs and the Laplace equation with discontinuity solutions are explored clearly, to display the significance of the proposed algorithms: this paper strengthens the first kind BIEs and its engineering applications.
