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The solutions of boundary value problems defined on cracked domains are usually non-smooth in the surroundings of the crack. In this work, we formulate the elasticity problem of a body with such geometric characteristic in a number of equivalent variational alternatives and show that we can take advantage of the theory of discontinuous finite elements in order to approximate its solution in an interesting way at little higher programming cost in comparison with the classical Galerkin method. The idea consists in splitting the global domain into a number of regions in which local mesh refinements are undertaken independently, producing irregular meshes with non-matching elements that are suitable to be used in discontinuous finite element methods. This strategy seems to be attractive to be employed in situations that we know in advance where the critical regions of the domain are located as well as in adaptive …