A posteriori error estimate for the symmetric coupling of finite elements and boundary elements

In this note we study a posteriori error estimates for a model problem in the symmetric coupling of boundary element and finite elements methods. Emphasis is on the use of the Poincaré-Steklov operator and its discretization which are analyzed in general for both a priori and a posteriori error estimates. Combining arguments from [6] and [9, 10] we refine the a posteriori error estimate obtained in [9, 10]. For quasi-uniform meshes on the boundary, we prove some inequality of a reverse type using techniques from [5] and [36]. This indicates efficiency of the new estimate as illustrated in a numerical example.