Operator decomposition methods are an attractive solution strategy for computing complex phenomena involving multiple physical processes, multiple scales or multiple domains. The general strategy is to decompose the problem into components involving simpler physics over a relatively limited range of scales, and then to seek the solution of the entire system through an iterative procedure involving solutions of the individual components. We construct an operator decomposition finite element method for a conjugate heat transfer problem consisting of a fluid and a solid coupled through a common boundary. Accurate a posteriori error estimates are then developed to account for both local discretization errors and the transfer of error between fluid and solid domains. These estimates can be used to guide adaptive mesh refinement. We show that the order of convergence of the operator decomposition finite element method is limited by the accuracy of the transferred gradient information, and demonstrate how a simple boundary flux recovery method can be used to regain the optimal order of accuracy in an efficient manner. This is joint work with Don Estep and Tim Wildey.