Efficient spectral method for the fourth order elliptic equation with a variable coefficient on the unit disc

In this paper, an efficient spectral-Galerkin method is proposed to solve the fourth order elliptic equation with a variable coefficient on the unit disc. The efficiency of the method is rest with the use of properly designed orthogonal polynomials as basis functions, which preserve the block-diagonal matrix structure of the discretized system with a constant coefficient and also work well with the general variable coefficient. Further, the three-term recurrence relation for orthogonal polynomials on the unit disc is explored to reduce the computational complexity of assembling the mass matrix. Then optimal error estimates are established. Finally, numerical experiments are carried out to illustrate the effectiveness of the proposed spectral method.