I've not used umbraco forms myself, but something like this should work

$(document).on("submit","#IDOFYOURFORM", function(event){
    event.preventDefault();
    var formValues= $(this).serialize();

    $.ajax({
        url: $(this).attr("action"),
        type: "POST",
        data: formValues
        , success: function (data) {
            if (data) {
                //this is whare you update your page.
                $("#form-outer-cases").html(data);
            } else {
                console.log('error: no data returned');
            }
        }, error: function (xhr) {
            console.log('Error: ', xhr.status + ' (' + xhr.statusText + ')');
        }
    }); 

});