If you're referring to Christian faiths that use the term, there isn't an official abbreviation for the reason that "Elder" is not used as a personal title.
Some Christian denominations use the term "elder" because the biblical records of first-century congregations say that "older men" took the lead and did not accept "honorific titles". So whereas some denominations do say, for example, "the Reverend Smith" or "Father Jones", elders are not normally addressed using the title. They may be known as "an elder" in their faith, but in secular life use titles like "Mr" and would be called by their familiar name among people they know.
Also worth noting is that countries which recognise titles like "reverend" in place of other salutations do so only if the person is ordained as a minister of a recognised religion, but. Christian elders are not ordained into that position as such. "Elder" is relative term - that is, it recognises experience. Some such faiths consider all of their members to be ordained ministers, and elders are just appointed for their experience.
I quickly researched the use of the term in the Church of Latter-Day Saints (Mormons), as I have seen their missionaries wearing name-badges that say, for example, "Elder Jones" on them. According to the Wikipedia entry on the church "The title "Elder" is not normally used as a personal title (e.g., Elder Evans, Elder Johnson), except by the LDS Church's general authorities, area seventies, and full-time male missionaries." So apparently their use of "elder" as a title is for internal use only and they do not use the term in secular life.