The title of the poem creates many different expectations: in fact there are many reasons not to mourn in life, and I don't know which one the poet may exploit: the poem will contain one of them.
Analyzing the structure, you will find it is a song organized into nine quatrains, so it should recall music with its sound effects, given by an alternate rhyme.
Probably the poem was written for Donne’s wife in occasion of his journey to the continent. Donne really travelled very much in his life. He invites her not to mourn for his departure.
Donne used a complex language. The poem can be considered an argumentative poem.
The poet makes a parallelism between men who die and the separation of two lovers. The only difference is that a dead man cannot come back home as does a lover. Donne uses two hyperboles, tear-floods and sigh-tempests, to invite his wife not to become sad because his departure is not a tragedy.
The third quatrain can be interpreted as a criticism to superstition and an invitation to face problems rationally.
In stanza four Donne resumes the theme of love. The “Dull sublunary lovers” are human beings who base their love only on body contact. In this case the absence will dissolve the bond of the lovers like death dissolve men’s bodies.
Quatrains five and six provide a description of poet’s and his wife’s love.

The last three quatrains form a metaphor. The lovers are compared to the two “stiff twin compasses”: only if one fix in the centre (in this case his wife) the other can draw a perfect circle and “makes me end where I begun”. Donne’s wife is his firm point. During the Elizabethan Age the circle was considered a symbol for perfection; this is the reason why Donne associates the circle to his love.
Men’s death and the separation of two lovers is not an ending but the beginning of a new cycle.