Treatment of Dirichlet-type boundary conditions in the spline-based wavelet Galerkin method employing multiple point constraints
Treatment of Dirichlet-type boundary conditions in the spline-based wavelet Galerkin method employing multiple point constraints
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DOI：
10.1016/j.apm.2016.11.018
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发表时间：
2017-03
影响因子：
5
通讯作者：
S. Sannomaru;Satoyuki Tanaka;K. Yoshida;T. Bui;S. Okazawa;S. Hagihara
中科院分区：
文献类型：
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作者：
S. Sannomaru;Satoyuki Tanaka;K. Yoshida;T. Bui;S. Okazawa;S. Hagihara
小波方法已被广泛地应用于求解偏微分方程的各种数值方法中，并与之相结合。然而，小波函数不满足Kronecker Delta函数的性质，因此需要特殊的处理方法来施加Dirichlet型边界条件。本文利用多点约束(MPC)和自适应性的优点，提出了一种新的基于样条基的小波Galerkin方法(WGM)的本质边界条件(BCS)处理技术。基函数采用线性B-样条尺度函数和多层小波函数。讨论了本方法的有效性，特别是对预测控制的适用性进行了研究。在该方法中，基于小波基函数沿基本边界条件的捆绑关系，建立了预测控制方程。基于预测控制方程对刚度阵进行退化，以施加边界约束。数值实现简单，在线性方程组中不需要额外的自由度。通过包括适应性分析在内的一些有代表性的数值例子，说明了本公式在处理世界地质模型中的边界条件的精确度。
The wavelet methods have been extensively adopted and integrated in various numerical methods to solve partial differential equations. The wavelet functions, however, do not satisfy the Kronecker delta function properties, special treatment methods for imposing the Dirichlet-type boundary conditions are thus required. It motivates us to present in this paper a novel treatment technique for the essential boundary conditions (BCs) in the spline-based wavelet Galerkin method (WGM), taking the advantages of the multiple point constraints (MPCs) and adaptivity. The linear B-spline scaling function and multilevel wavelet functions are employed as basis functions. The effectiveness of the present method is addressed, and in particular the applicability of the MPCs is also investigated. In the proposed technique, MPC equations based on the tying relations of the wavelet basis functions along the essential BCs are developed. The stiffness matrix is degenerated based on the MPC equations to impose the BCs. The numerical implementation is simple, and no additional degrees of freedom are needed in the system of linear equations. The accuracy of the present formulation in treating the BCs in the WGM is high, which is illustrated through a number of representative numerical examples including an adaptive analysis.