It's trivial to use a passive cable to pass DisplayPort output from a video source to a DVI input on a monitor (I have one myself to connect a laptop docking station to a KVM switch to a DVI monitor input). However, these cables are NOT bidirectional, and you need an active converter (such as that suggested by PJD1) to achieve the necessary signal conversion from DVI output to DisplayPort input.
Still, it's cheaper than a new graphics card, or monitor! Although I have no great faith in these active devices working reliably over long periods.
SD