Storing references to DOM elements

In a particular script I'm writing, I have a number of objects which are linked to some DOM Elements. Given that each element has a unique id, should each object keep just the element's id (and use document.getElementById each time), or store the element in a property?

Here's a simplified example of what I mean:

function myThing(elId) {
    this.elId = elId;
}
myThing.prototype.getElValue = function() {
    return document.getElementById(this.elId).nodeValue;
};

// -- vs -- //

function myThing(elId) {
    this.el = document.getElementById(elId);
}
mything.prototype.getElValue = function() {
    return this.el.nodeValue;
};

Does it make any difference? Are there any performance issues I should know about?