Hybrid multicore processors (HMPs) are poised to dominate the landscape of the next generation of computing on the desktop as well as on exascale systems. HMPs consist of general purpose GPU cores along with specialized cores and are expected to provide benefits to a wide spectrum of applications at significantly lower energy requirements per FLOP (FLoating-point Operations per Second).In this paper, we describe a comprehensive strategy for efficiently implementing a key kernel of spectral solvers on GPUs and HMPs consisting of CPU and GPUs. Our implementations represent significant computational improvements of the kernel on GPU. We also provide load balancing strategies for a combination of CPU and GPU cores. We show that depending upon whether performance or energy optimization is critical, separate load balancing strategies should be used on HMPs. We provide performance, power and energy evaluation as well as modeling and validation for a variety of CPU–GPU core combinations.