|
| 摘要: |
| 本文提出了椭圆域上二阶/四阶变系数问题的一种有效的谱Galerkin逼近.首先,我们将原问题化为极坐标下的等价形式,并建立其弱形式及相应的离散格式.其次,针对二阶情形,我们证明了弱解和逼近解的存在唯一性及它们之间的误差估计.另外,根据极条件和勒让得多项式的正交性,我们构造了一组有效的径向基函数,并在θ方向作截断的傅立叶展开,推导了离散格式等价的矩阵形式.最后,我们给出了大量的数值算例,数值结果表明了我们算法的收敛性和谱精度. |
| 关键词: 二阶/四阶问题 谱Galerkin方法 误差分析 椭圆区域 |
| DOI： |
| 分类号:O241.82 |
| 基金项目:国家自然科学基金资助(12061023),贵州师范大学学术新苗基金资助(黔师新苗[2021]A04号). |
|
| EFFECTIVE SPECTRAL GALERKIN APPROXIMATION FOR SECOND-ORDER/FOURTH-ORDER VARIABLE COEFFICIENT PROBLEMS IN AN ELLIPTIC DOMAIN |
|
TIAN Xiao-hong,AN Jing
|
| Abstract: |
| 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 . |
| Key words: Second-order/fourth-order problems spectral Galerkin method error analysis elliptic domain |
