Explain the role of hamartia and catharsis in tragedy

According to Wikipedia.com, hamartia
is a term that was developed by Aristotle and:

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...can simply be seen as a character’s flaw or
error...hamartia is the tragic flaw of the protagonist in a given
tragedy.

The word hamartia is rooted in the notion of
missing the mark (hamartanein) and covers a broad spectrum that includes accident and
mistake, as well as wrongdoing, error, or
sin.

Shakespearean tragedies
come to mind, such as Hamlet and Macbeth.
Shakespeare's tragic heroes all had a tragic [character] flaw. For Hamlet it was
indecision. For Macbeth it was vaulting
("blind")
ambition.

EncyclopediaBritannicaeb.com
notes:

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....misfortune is not brought about by villainy
but by some “error of judgment” (hamartia). This imperfection later came to be
interpreted as a moral flaw, such as Othello’s jealousy or Hamlet’s
irresolution...

The basis for
understanding hamartia, then, is that a mistake is made, but it is due more to an error
in judgment, or, more typically perhaps, a flaw in one's character, rather than
springing from malicious intent. It was something created by Aristotle. Catharsis is
also something connected to the Greek, and it also was used by
Aristotle.

Catharis means "cleansing" or "purging," but it
is not necessarily referred to literally. One may experience a catharsis by experiencing
a bout of prolonged weeping. Screaming or yelling, or even unburdening one's guilt can
be cathartic. The idea is that a weight is lifted off of one's
heart or soul.

The emphasis of "catharsis" as an emotional
response was introduced by Aristotle as well.

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The Greek philosopher Aristotle was the first to
use the term catharsis with reference to the emotions—in his
work Poetics.  In that context, it refers to a sensation or
literary effect that, ideally, would either be experienced by the characters in a play,
or be wrought upon the audience at the conclusion of a tragedy; namely, the release of
pent-up emotion or
energy.

Therefore, in a
tragedy, hamartia refers to a hero's tragic flaw, which drives him
to do things that not only affect those around him, but ultimately his own fate as
well.

Catharsis is something that can also be seen in a
tragedy. For instance, when Claudius and Macbeth, two of Shakespeare's great villains,
die (in Hamlet and Macbeth, respectively), it
may be cathartic for the audience, seeing such heinous criminals punished for their
horrific deeds. When Hamlet kills Claudius, we may assume it is cathartic for him as
well, although his tragic flaw (harmatia) has already sealed his own
fate.