High-Order Numerical Approximation and Error Estimation for Fourth-Order Equation in Complex Cylindrical Domain

Abstract

In this paper, we introduce a high-order numerical method for solving fourth-order equations in an elliptical cylindrical region. Initially, we employ an elliptical cylindrical coordinate transformation to reformulate the fourth-order equation as an equivalent second-order coupled system in the new coordinate system. To overcome the singularity introduced by coordinate transformation, an essential pole condition is derived. The weak form and its discretization are also established. Furthermore, the well-posedness of both the weak solution and its approximate solution have been theoretically investigated. By introducing novel projection operators and demonstrating their approximation properties, we prove the error estimates in conjunction with the approximation results of Fourier series. Finally, several numerical examples are provided to validate the convergence and high accuracy of our proposed schemes.