Coupling techniques of Trefftz methods - Huang, Hung-Tsai; Li, Zi-Cai; Cheng, Alexander H.-D. (Computer Assisted Methods in Engineering and Science, 2022)
The Trefftz method pioneered by Trefftz [71] in 1926 is described as follows: The particular solutions or the fundamental solutions are chosen, a linear combination of those functions is regarded as an approximate solution of partial differential equations (PDEs), and their expansion coefficients are sought by satisfying the interior and exterior boundary conditions. When the solution domain is not rectangular or sectors, the piecewise particular solutions may be chosen in different subdomains, and some coupling techniques must be employed along their interior boundary conditions. In Li et al. [49], the collocation method is used for the Trefftz method, to lead to the collocation Trefftz method (i.e., the indirect Trefftz method). In this paper, we will also discuss other four coupling techniques: (1) the simplified hybrid techniques, (2) the hybrid plus penalty techniques, (3) the Lagrange multiplier techniques for the direct Trefftz method, and (4) the hybrid Trefftz method of Jirousek [23] and Qin [62]. Error bounds are derived in detail for these four couplings, to achieve exponential convergence rates. Numerical experiments are carried out, and comparisons are also made.