Summary Hydraulic fracture (HF) modeling is a multiscale and multiphysics problem. It should capture various effects, including those of in-situ stresses, poroelasticity, and reservoir heterogeneities at different length scales. A peridynamics (PD)-based hydraulic fracturing simulator has been demonstrated to reproduce this physics accurately. However, accounting for such details leads to a reduction in computational speed. In this paper, we present a novel coupling of the PD-based simulator with numerically efficient finite element methods (FEMs) and finite volume methods (FVMs) to achieve a significant improvement in computational performance. Unlike classical methods, such as FEM and FVM that solve differential equations, PD uses an integral formulation to circumvent the undefined spatial derivatives at crack tips. We implemented four novel coupling schemes of our PD-based simulator with FEM and FVM: static PD region scheme, dynamic PD region scheme, adaptive mesh refinement scheme, and dynamic mesh coarsening scheme. PD equations are solved using a refined mesh close to the fracture, whereas FE/FV equations are solved using a progressively coarser mesh away from the fracture. As the fracture grows, a dynamic conversion of FE/FV cells to PD nodes and adaptive mesh refinement are incorporated. To improve the performance further, the dynamic mesh coarsening scheme additionally converts the fine PD nodes back to coarse FE/FV cells as the HF grows in length. The coupling schemes are verified against the Kristianovich-Geertsma-de Klerk (KGD) fracture propagation problem. No spurious behavior is observed near the transition between PD and FE/FV regions. In the first three coupling schemes, the computational runtime for single fracture propagation is reduced by up to 10, 20, and 50 times, respectively, compared to a pure PD model. Laboratory experiments on the interaction of an HF with a natural fracture (NF) are revisited. The model captures complex fracture behavior, such as turning in the case of low stress contrast and low angle of interaction, kinking for higher stress contrast or higher angle of interaction, and fracture crossing for near-orthogonal NFs. Moreover, several previously reported phenomena, including fracture propagation at an angle to the principal stress directions, competing fracture growth from multiple closely spaced clusters, and interaction with layers of varying mechanical properties are successfully modeled. Thus, the coupling of PD with FEM and FVM offers an innovative and fundamentally comprehensive solution to alleviate the high computational costs typically associated with the pure PD-based hydraulic fracturing simulations. At the same time, these coupling schemes retain the versatility of the nonlocal PD formulation at modeling the evolution of arbitrary material damage, commonly observed during HF propagation in complex heterogeneous reservoirs.