Convergence Enhancement for Embedded Domain Decomposition Method Through Spectral Radius Reduction

The embedded domain decomposition method (DDM) has emerged as an innovative DDM variant for electromagnetic simulations, offering enhanced flexibility in geometrical modeling and mesh generation. While previous studies have shown rapid convergence for this method, our recent applications have uncovered certain convergence challenges. Targeting these convergence issues, this work introduces a preconditioner by constructing an approximate inverse to the system matrix of embedded DDM. The preconditioner is built upon two major ingredients: randomized sampling and spectral radius reduction. Through randomized sampling of system equations, we efficiently extract essential information from the DDM matrix without the need for explicit entry evaluations. These samples are then progressively compressed in an adaptive manner combined with spectral radius estimations. Numerical experiments demonstrate the impact of spectral radius on system convergence and the effectiveness of numerical compression in optimizing system eigenspectrum. Subsequent validation through numerical examples showcases the preconditioner's capability in ensuring robust convergence and improving stability of embedded DDM.