The idea behind using the reduced basis element method for the thermal fin problem is threefold: (i) to decompose the geometry \(\Omega\) into elementary bricks that resemble a fixed reference shape \(\widehat \Omega\), (ii) to express the approximate numerical solution within each particular brick as a linear combination of precomputed solutions for similar parts, (iii) to glue together the solutions on the individual parts by using Lagrange multipliers. An a posteriori error analysis is presented.