Spectral Method And Error Analysis Based On Dimensionality Reduction Scheme For The Fourth-order Steklov Problem And Maxwell Eigenvalue Problem In Polar Geometries

Based on the global high-order polynomial approximation and dimensionality reduction techniques,an efficient spectral method algorithm is proposed to solve the fourth-order Steklov source problem and eigenvalue problem in polar geometries and the Maxwell eigenvalue problem in spherical domain,the rigorous proof of error estimations are also established.For the fourth-order Steklov source problem in the circular domain,we first used the polar coordinate transformations and the orthogonality of Fourier basis functions to decompose the original problem into a set of equivalent one-dimensional fourth-order problems.Secondly,in order to eliminate the singularities introduced by the polar coordinate transformation,we derived the corresponding essential pole conditions and defined the appropriate weighted Sobolev space,then we deduced the corresponding weak form and discrete scheme accordingly.Thirdly,by using Lax-Milgram lemma and the properties of the Sobolev space projection operators,we proved the existence and uniqueness of weak solutions and approximated solutions and the error estimations between them,respectively.Fourthly,we constructed a set of basis functions which satisfy the boundary conditions and pole conditions,and verified the correctness of the theoretical analysis and the effectiveness of the algorithm by the numerical experiments.Finally,we also applied the proposed algorithm to the fourth-order Steklov eigenvalue problem in the circular domain.In addition,by using the spherical coordinate transformation and the orthogonality of spherical harmonics,we derived the dimensionality reduction scheme of the fourth-order Steklov eigenvalue problem in the spherical domain,the corresponding spectral method algorithm and numerical experiments are proposed accordingly.For the Maxwell eigenvalue problem in spherical domain,by using the spherical coordinate transform and orthogonality of the vector spherical harmonics,we decomposed the original problem into two independent onedimensional eigenvalue problems,the so-called TE mode and TM mode.For TE mode,we established its weak form and discrete scheme directly,and proved the error estimations of approximate eigenvalues and eigenfunctions.For the TM mode,we derived the weak form and its discrete scheme based on the parameterized method to filter out spurious eigenvalues.In addition,since TM mode is a coupled problem,we also constructed a set of Legendre vector basis functions to solve the coupled discrete system numerically.Finally,we conducted a large number of numerical experiments,which verified the correctness of the theoretical analysis and the effectiveness of the algorithm.