The transition towards heterogeneous or GPU-dominated hardware in high-performance computing (HPC) infrastructure has highlighted the need for hardware-extensible implementations of HPC applications in scientific computing and AI. Ensuring that these implementations are optimized for the specific hardware target poses a significant challenge, as the optimization process requires specific hardware-level knowledge that is not only outside the scientific domain but also varies across different hardware types and vendors. In this talk, we present NektarIR, a domain-specific compiler for high-order finite element operations on CPU and GPU hardwares. Using the MLIR project, we have created an intermediate representation of a high-level abstraction of the operations that is rewritten and lowered through a series of passes to optimized and hardware-specific kernels. These kernels are just-in-time (JIT) compiled for the specific hardware and architecture using LLVM and can be used in the construction of spectral/hp element solvers for computational fluid dynamics. We also present benchmarks for the overhead introduced by our compiler pipeline and a run-time performance comparison of our kernels against the Nektar++ spectral/$hp$ element framework.