We develop a spectral element method (SEM) for simulating dynamic rupture on rate and state faults and use it to study how the rupture is affected by a shallow fault region of steady state velocity-strengthening friction. Our comparison of the developed SEM and a spectral boundary integral method (BIM) for an antiplane (two-dimensional) test problem shows that for the finest resolution that we use, the two methods produce virtually identical solutions, with negligible differences in rupture arrival times and peak slip velocities (less than 0.05% of their absolute values). The convergence with grid reduction of the developed SEM is comparable to that of BIM. We also use the test problem to compare numerical resolution required for different state evolution laws and for linear slip-weakening friction. Using our three-dimensional implementation of the methodology, we find that a shallow velocity-strengthening fault region can significantly alter dynamic rupture and ground motion. The velocity-strengthening region suppresses supershear propagation at the free surface occurring in the absence of such region, which could explain the lack of universally observed supershear rupture near the free surface. In addition, the velocity-strengthening region promotes faster falloff of slip velocity behind the rupture front and decreases final slip throughout the entire fault, causing a smaller average stress drop. The slip decrease is largest in the shallow parts of the fault, resulting in a depth profile of slip qualitatively consistent with observations of shallow coseismic slip deficit. The shallow velocity-strengthening region also reduces the amplification of strong ground motion due to a low-velocity bulk structure.