Summary: We define an adaptive version of a recently introduced finite element method for numerical treatment of elliptic partial differential equations (PDEs) defined on surfaces. The method makes use of a (standard) outer volume mesh to discretize an equation on a two-dimensional surface embedded in \(\mathbb{R}^3\). Extension of the equation from the surface is avoided, but the number of degrees of freedom is optimal in the sense that it is comparable to methods in which the surface is meshed directly.
In previous work it was proved that the method exhibits optimal order of convergence for an elliptic surface PDE if the volume mesh is uniformly refined. In this paper, we extend the method and develop an a posteriori error analysis which admits adaptively refined meshes. The reliability of a residual type a posteriori error estimator is proved and both reliability and efficiency of the estimator are studied numerically in a series of experiments. A simple adaptive refinement strategy based on the error estimator is numerically demonstrated to provide optimal convergence rate in the \(H^1\) norm for solutions with point singularities.