<form method="GET" action="/popup-teste" id="ID_DO_FORMULARIO" onsubmit="return atualizar_div()">
    <button type="submit>OK!</button>
    <input name="teste" id="input1">
</form>

<script type="text/javascript">

function atualizar_div() {
        var frm = $('#ID_DO_FORMULARIO');
        $.ajax({
            type: frm.attr('method'),
            url: frm.attr('action'),
            data: frm.serialize(),
            success: function (data) {
                $("#DIV_PARA_ATUALIZAR").html(data);
            },
            error: function(data) {
                $("#DIV_PARA_ATUALIZAR").html("Algo deu errado!");
            }
        });
        return false;
    }
</script>