In this talk I will discuss the technical and design challenges of developing high-order finite element flow solvers which can efficiently take advantage of GPU accelerators present in current and next-generation Leadership Computing Facilities. I will detail how despite high-order methods being well-suited for approximating smooth flows, the GPU performance of operators and linear solves degrades as the polynomial degree increases. I will describe how we can construct empirically determined roof-line models to guide the optimization of GPU compute kernels in order to maximize performance, and how choosing a basis which gives sparser operators can yield even greater performance gains. I will then discuss performance of GPU accelerated preconditioners for high-order elliptic finite element problems and describe a new low-order preconditioning technique for triangle elements where the low-order and high-order interpolation nodes do not necessarily coincide, but are connected via a least squares projection. I will present some computational results obtained in three dimensional incompressible flows and scaling studies.