On the Approximate Component Mode Synthesis method for the approximation of the Helmholtz equation

This dissertation introduces and investigates an extension of the Approximate Component Mode Synthesis (ACMS) method, originally developed for elliptic partial differential equations, to the two-dimensional heterogeneous Helmholtz equation. The ACMS method is a multiscale approach that employs a non-overlapping domain decomposition to split the numerical solution into two independent components: local Helmholtz problems, which can be solved in parallel, and a global interface problem, addressed using locally supported basis functions constructed from eigenmodes and suitable extensions. For a fixed decomposition, the method achieves algebraic convergence, and in some cases even super-algebraic convergence, without requiring oversampling.

Next, the thesis presents an implementation of the ACMS method using high-order hp-finite elements within the open-source package NGSolve, enabling more accurate eigenfunction approximations and improved computational efficiency. Numerical experiments show that high-order elements significantly enhance the convergence rate of the method, particularly at higher wavenumbers, and effectively mitigate the pollution effect commonly observed with linear elements. The implementation demonstrates both computational efficiency and straightforward parallelization. The number of edge modes required for convergence scales linearly with the wavenumber, with rapid initial error decay and low sensitivity to the domain decomposition. The method is applied to large photonic crystals with defects, where system sizes remain moderate, allowing the use of sparse direct solvers.

Finally, the ACMS framework is extended to cases with discontinuous material parameters aligned with the domain decomposition interfaces. In these lower-regularity settings, refined error estimates are derived and numerically validated, establishing convergence rates even in the presence of singular points caused by corners in the material coefficients. Numerical results further show that, although not covered by the theoretical bounds, higher convergence rates are achieved locally away from the critical regions.