Elliptic curve scalar multiplication (ECSM) stands as a crucial sub-block in elliptic curve cryptography, which represents the most widely-used pre-quantum public key cryptography. Hardware constructions of cryptographic systems utilizing ECSM have been subject to permanent or transient errors. In cryptographic systems, it is important to validate the correctness of the underlying computation performed on hardware or software to identify such errors. In this paper, we present new fault detection schemes in window method scalar multiplication, which, to the best of our knowledge, has not been previously investigated. Our approach involves introducing refined algorithms and implementations that can effectively counter both permanent and transient errors. We assess this by simulating a fault model, ensuring that the evaluations conducted reflect the obtained results. As a result, we achieve a significantly extensive coverage of errors. Lastly, we benchmark our proposed error detection scheme on ARMv8 and FPGA to demonstrate the implementation and resource overhead. On Cortex-A72 processors, we maintain a clock cycle overhead of under 3%. Additionally, when implementing our error detection method on different FPGAs including Zynq Ultrascale+, Artix-7, and Kintex Ultrascale+, we achieve comparable throughput while introducing a mere 2% increase in area compared to the original hardware implementations.