An high-order numerical method based on Legendre polynomial approximation for fourth-order eigenvalue problem in cylinder domain

英文关键词:Fourth-order equation, decoupled reduced-dimension formulation, Legendre-Galerkin, error estimate, cylinder domain

中文摘要:

      

英文摘要:

      In this work, an efficient spectral method is proposed to solve the fourth-order eigenvalue problem in cylinder domain. Firstly, the key point of this method is to decompose the original model into a kind of decoupled two-dimensional eigenvalue problem by cylindrical coordinate transformation and Fourier series expansion, and deduce the crucial essential pole conditions. Secondly, we define a kind of weighted Sobolev spaces, and establish a suitable variational formula and its discrete form for each two-dimensional eigenvalue problem. Furthermore, we derive the equivalent operator formulas and obtain some prior error estimates of spectral theory of compact operators. More importantly, we further obtained error estimates for approximating eigenvalues and eigenfunctions by using two newly constructed projection operators. Finally, some numerical experiments are performed to validate our theoretical results and algorithm.