I assume this is .gpx files that don't already include altitude information.
Otherwise you could just use Excel or GPS Altitude Chart (requires Java). :-|
The elevation info from the GPS is often dodgy, especially in the period shortly after turning on the GPS and it slowly acquires the fix, or when you've been static for a while, or if you've been climbing up under tree cover, etc.
The GPX file from this weekends ride showed me climbing/descending a total of 50m whilst the bike was leant against a post (with an unobscured view of the sky) for 20 minutes at an Audax control.
This is where some smoothing needs to be done, and the DEM data should help with that.