arXiv:2610.07082v1 Announce Type: new Abstract: We propose the safeguarded Anderson-accelerated reflection (SAAR) method, a two-level iterative method for the continuous Sylvester matrix equation $AX+XB=C$. The core of the method is a fixed-point reformulation: at each outer iteration, the associated linear matrix equation is decomposed into independent column-wise linear systems, which are solved approximately by a reflection iteration. For nonsingular \(A\) and \(n\ge 2\), we prove the convergence of the reflection inner iteration by a spectral-radius argument and extend the result to the matrix equation setting. Since only finitely many inner iterations are used in practice, the overall scheme is analyzed as an inexact fixed-point iteration, and a sufficient condition for linear convergence of the unaccelerated inexact iteration is established in the Frobenius norm. On top of this scheme, SAAR incorporates Anderson acceleration into the outer iteration, while a residual-based safeguard rejects Anderson candidates whose Sylvester residual is larger than that of the standard inexact reflection iterate, thereby enhancing numerical stability. Numerical experiments indicate that SAAR substantially reduces the number of outer iterations and often the CPU time, and it improves empirical robustness with respect to the inner tolerance for the tested problems.