If you want both stdout and stderr, use:
( application_to_run 2>&1 ) | grep FATAL
If you want both stderr alone, you can use:
( application_to_run 2>&1 >/dev/null ) | grep FATAL
The first sends all output destined for file handle 2 (stderr) to file handle 1 (stdout), then pipes that through grep. The second does the same but also sends stdout to the bit bucket. This will work since redirection is a positional thing. First, stderr is redirected to the current stdout, then stdout is redirected to /dev/null.