Improved Numerical Robustness of the X‐FFT Solver via Internal Scaling

ABSTRACT The recently introduced X‐FFT solver improves the spatial accuracy of FFT‐based homogenization methods for two‐dimensional thermal homogenization problems without compromising their numerical efficiency. Through the use of an X‐FEM discretization, optimal error convergence rates of the discretization error are achieved, and the developed X‐FFT preconditioner guarantees a mesh‐independent upper bound on the condition number. In the work at hand, we introduce an internal scaling of the enriched shape functions that prevents numerical instabilities and simplifies the X‐FFT preconditioner. Our numerical studies demonstrate that the accuracy and the efficiency of the X‐FFT solver remain unaffected by the internal scaling.