An elite coevolutionary mayfly algorithm (ECMA) was proposed to resolve the small population diversity and the poor optimization performance of the mayfly algorithm. Firstly, male mayflies were divided into elite and ordinary members based on their fitness performance, then the elite individuals learned from itself to maintain the population diversity and achieve high-level global search, while the ordinary individuals flew toward a unified target for local development to improve the convergence speed of ECMA. Secondly, the position update of female mayflies was improved based on the marriage market theory, thus enhancing the optimization performance of ECMA. Thirdly, a new adaptive gravity coefficient was introduced to establish a balance between the global search and the local development to improve the convergence accuracy of ECMA. Finally, a jump-out strategy of Levy flight was introduced to avoid ECMA falling into a local optimum. Based on 20 benchmark test functions and CEC2019 test functions, the simulation optimization analysis of the algorithm was carried out. Compared with the mayfly algorithm and other excellent swarm intelligence algorithms, ECMA has greatly improved the optimization accuracy, convergence speed and stability.