四阶问题基于降维格式的混合有限元法

A Mixed Finite Element Method Based on A Dimensionality Reduction Scheme for Fourth-order Problems

  • 摘要: 为了有效地数值求解球域上的四阶问题,文章提出了球域上四阶问题基于降维格式的一种混合有限元法:首先,利用球坐标变换和球谐函数的正交性,将原始问题分解为一系列解耦的一维四阶问题,通过引入一个辅助的中间变量,进一步将其化为一个等价的二阶耦合系统。其次,定义了一类乘积型的带权Sobolev空间,对每个二阶耦合系统建立一种混合变分形式及其离散格式,并从理论上证明了弱解及其逼近解的存在唯一性以及它们之间的误差估计。另外,详细描述了基函数的构造与离散变分形式等价的矩阵形式。最后,给出了一些数值算例,数值结果表明了离散变分形式的有效性和收敛性。

     

    Abstract: To numerically solve the fourth-order problem on the spherical domain effectively, in this article, a mixed finite element method based on a dimension reduction scheme is proposed for fourth-order problems in sphe- rical domain. Initially, by employing a spherical coordinate transformation and the orthogonality of spherical harmo-nics, the original problem is split into a series of decoupled one-dimensional fourth-order problems. Through the introduction of an auxiliary intermediate variable, each fourth-order problem is further transformed into an equivalent second-order coupled system. Additionally, a class of product-type weighted Sobolev spaces is defined, a mixed variational form and its discrete scheme are established for each second-order coupled system, and theoretically demonstrate the existence and uniqueness of the weak and approximate solutions and the error estimates between them. Furthermore, the construction of basis functions and the matrix form equivalent to the discrete variational scheme are described in detail. To validate the effectiveness and convergence of the discrete variational scheme, several numerical examples are presented.

     

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