High‐Order Numerical Approximation Based on the Ciarlet‐Raviart Mixed Scheme for Fourth‐Order Problem in a Complex Domain

In this paper, we propose a high‐order numerical approximation based on the Ciarlet‐Raviart mixed scheme for a fourth‐order problem in a complex domain. Initially, we introduce auxiliary variables to decompose the fourth‐order problem into a coupled system of second‐order equations. Subsequently, we further transform it into an equivalent form in polar coordinates. Then, by defining a weighted two‐dimensional Sobolev space, we establish the mixed variational form of the equivalent form and its discretization. We conduct a theoretical demonstration of the existence and uniqueness of weak solutions and their corresponding approximate solutions, along with the error estimates between them. It is particularly worth mentioning that we have broadened the application scope of our algorithm even further, with the aim of tackling the challenging fourth‐order Steklov eigenvalue problem in complex regions. Finally, we validate the effectiveness and spectral accuracy of the algorithm through numerical examples.