Abstract:
To numerically solve the fourth-order problem on the spherical domain effectively, in this article, a mixed finite element method based on a dimension reduction scheme is proposed for fourth-order problems in sphe- rical domain. Initially, by employing a spherical coordinate transformation and the orthogonality of spherical harmo-nics, the original problem is split into a series of decoupled one-dimensional fourth-order problems. Through the introduction of an auxiliary intermediate variable, each fourth-order problem is further transformed into an equivalent second-order coupled system. Additionally, a class of product-type weighted Sobolev spaces is defined, a mixed variational form and its discrete scheme are established for each second-order coupled system, and theoretically demonstrate the existence and uniqueness of the weak and approximate solutions and the error estimates between them. Furthermore, the construction of basis functions and the matrix form equivalent to the discrete variational scheme are described in detail. To validate the effectiveness and convergence of the discrete variational scheme, several numerical examples are presented.