The Trefftz method for the Helmholtz equation with degeneracy

The Trefftz method (TM) [E. Trefftz, Ein Gegenstuck zum Ritz'schen Verfahren, in: Proc. 2nd Ind. Congr. Appl. Mech., Zurich, 1926, pp. 131–137] (i.e., Boundary approximation method) is developed to solve the Helmholtz equation, Δu+k2u=0, where k2 is not exactly equal (but may be very close) to an eigenvalue of the operator −Δ. Piecewise particular solutions are chosen and then matched together in order to satisfy the exterior and interior boundary conditions. Error analysis is presented to estimate error bounds for the Helmholtz solutions in the entire solution domain. Let δ be the smallest relative distance between k2 and the eigenvalues of −Δ. We prove that the error asymptote of the solutions by the TM is as δ→0, which is called degeneracy in this paper. Such an asymptote has been verified by the numerical computations. We also explain why the exponential convergence rates of solutions can be obtained easily by splitting the solution domain into smaller subdomains. Numerical experiments of both smooth and singular solutions are provided, to support the TM algorithms and the error analysis made.