Dynamically create the javascript using Control.ClientID to determine the calculated ID of div.

document.getElementById('<%= DivControl.ClientID %>')

Or search for the element on the client side using the base ID as a search pattern. See here: A generic way to find ASP.NET ClientIDs with jQuery

I prefer the server side calculation, but if you don't do it often and/or your current design prohibits it, the client side way is a reasonable workaround.

you can check i the element exists by checking if it returns not null

if (document.getElementById('divId') != null) { /* do your stuff*/ }

in other words:

if (document.getElementById('divId')) { /* do your stuff*/ }

now you have edited you orginal question i got it.. i would do something like this:

var arrDivs = document.getElementsByTagName('div'),
    strDivName = "divId";

for (i=0;i<=arrDivs.length;i++){
    if( arrDivs[i].id.indexOf(strDivName) != -1) {
        alert("this is it")
    }
}

you can see a demo here: http://jsfiddle.net/pnHSw/2/

i think you could do it better with a regex.

But this is a pure JS way i don't know ASP.net

edit: i think Aristos solution is much cleaner :P

maybe you can use a descendent selector un css

<div id="wrapperControler">
    <controler id="controler"></controler>
</div>

wrapperControler controler{

 dosomething;

}