An Extended Virtual Element Method for the Shallow Shell Model

ABSTRACT In this work, we develop an extended virtual element discrete scheme for the shallow shell model proposed by Ciarlet and Miara. The shallow shell model is a fourth‐order variable coefficient shell, and when applying traditional and virtual element approximations, it becomes necessary to increase the degrees of freedom to ensure the computability of each term in the shallow shell equation. To address this, we introduce an extended discrete space with the minimum degrees of freedom by employing an projection operator that satisfies the orthogonality condition. For the same error magnitude, the number of degrees of freedom required in the extended virtual element method is significantly lower than that in the finite element method. Moreover, we obtain the existence and uniqueness of the virtual element discrete solution, and analyze the error estimation under the energy norm. Finally, numerical experiments on paraboloid, regular saddle surface, and irregular saddle surface on polygonal meshes verify that the discrete scheme is convergent and stable.