A process server went to Laura Dern's home on Saturday with divorce papers and demanded that she come to the door ... or else, sources tell TMZ.

We're told the couple's nanny came to the door and told the process server she would accept the divorce papers Ben Harper filed late Friday afternoon, but sources say the process server fired back that if Laura didn't come to the door he'd find her out in public and embarrass her.

A source connected with Ben claims the allegation that the process server issued such an ultimatum is "absolutely false."

The divorce -- a story TMZ broke Sunday -- shocked Laura, who didn't see it coming.

We're told Ben has not called Laura to discuss how to tell their two children about the sudden turn of events ... he just disappeared.

And, as we reported, although Ben claims he and Laura have been separated since January, the couple showed up together at Eddie Vedder's wedding late last month.

And, sources connected with the couple tell us Laura and Ben have been intimate as recently as last week.