Summary: This article concerns the numerical analysis and the error estimate of the biharmonic problem with homogeneous boundary conditions using the mortar spectral element method in domains with corners. Since the solution of this problem can be written as a sum of a regular part and known singular functions, we propose to use the algorithm by G. Strang and G. J. Fix [An analysis of the finite element methods. 2nd ed. Wellesley, MA: Wellesley-Cambridge Press (2008; Zbl 1171.65081)] for improving the order of the error.