For a two-dimensional model problem, the weak form for a partial differential equation is formulated on a partitioned domain and coupled via Lagrange multipliers – the continuous mortar. The discrete finite-element spaces are introduced, both in the domain and the mortar space, and a local a posteriori residual-based error estimator is developed. This is extended to the Crouzeix-Raviart element of lowest order, which is interpreted as a mortar element method. The paper closes with numerical examples of adaptively refined meshes.