If you don't want to use sessions, the only thing you can do is POST to the same page. Which IMO is the best solution anyway.
// form.php
<?php
if (!empty($_POST['submit'])) {
// validate
if ($allGood) {
// put data into database or whatever needs to be done
header('Location: nextpage.php');
exit;
}
}
?>
<form action="form.php">
<input name="foo" value="<?php if (!empty($_POST['foo'])) echo htmlentities($_POST['foo']); ?>">
...
</form>
This can be made more elegant, but you get the idea...