A Trefftz method for the 3D interior Dirichlet Helmholtz problem

In this paper, we develop a Trefftz method with Tikhonov regularization for the three-dimensional interior Dirichlet problem of the Helmholtz equation. The Trefftz approximationspace is constructed from normalized spherical Bessel functions and spherical harmonics, sothat every trial function satisfies the Helmholtz equation exactly in the computational domain.Using Runge approximation and spherical-wave expansions, we establish an approximationresult for the proposed method. We then derive an explicit lower bound for the smallestsingular value of the discrete boundary trace operator, revealing its dependence on thetruncation degree, the wavenumber, the domain geometry, and the distance of the squaredwavenumber from the Dirichlet spectrum. Based on this stability estimate, we analyzeTikhonov regularization and obtain quantitative error estimates for noisy boundary data.Numerical experiments demonstrate the approximation accuracy, the effectiveness of basisnormalization, the deterioration of stability near Dirichlet eigenvalues, and the stabilizingeffect of regularization.