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An integral equation-fast Fourier transformation (IE-FFT) with grid-robust higher order vector basis functions (BFs) are presented. A conformal mesh is required for traditional BFs based on common edges between adjacent elements. This is a very rigorous requirement for large and complicated electrical geometries. Instead of a conformal mesh, grid-robust higher-order vector BFs are used here. It maintains the flexibility of geometry modelling and reduces the number of the unknowns owing to the property of point BFs. An IE-FFT algorithm based on a flexible floating stencil topology is proposed to accelerate the solution of the IE. Comparing the IE-FFT with traditional RWG BFs, the present method has a much lower error of interpolation, and the matrix is also simpler to implement. Numerical results are presented to demonstrate the accuracy and efficiency of this method.