椭圆域上二阶/四阶变系数问题有效的谱Galerkin逼近OACSTPCD

In this paper,we propose an efficient spectral Galerkin approximation for second-order/fourth-order problems with variable coefficients in an elliptic domain.First,we convert the initial problem into an equivalent form in polar coordinates.Subsequently,we establish the weak form and corresponding discrete scheme.Secondly,we prove the existence and uniqueness of weak and approximate solutions,and we also offer error estimates for the second-order case.In addition,based on the polar condition and the orthogonality of Legendre polynomials,we construct a set of effective radial basis functions,perform a truncated Fourier expansion in the direction of θ,and derive the equivalent matrix form of the discrete scheme.Finally,we provide a large number of numerical examples,and the numerical results show the convergence and spectral accuracy of our algorithm.