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Treatment of Dirichlet-type boundary conditions in the spline-based wavelet Galerkin method employing multiple point constraints

机译:多点约束的基于样条的小波Galerkin方法处理狄利克雷型边界条件

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摘要

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.
机译:小波方法已被广泛采用并集成到各种数值方法中来求解偏微分方程。但是,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。它激发我们在本文中提出一种新颖的基于样条的小波Galerkin方法(WGM)中的基本边界条件(BCs)的处理技术,该技术利用了多点约束(MPC)和适应性的优点。线性B样条缩放函数和多级小波函数被用作基础函数。解决了本方法的有效性,并且特别地,还研究了MPC的适用性。在所提出的技术中,建立了基于小波基函数沿着基本BC的束缚关系的MPC方程。基于MPC方程对刚度矩阵进行简并以施加BC。数值实现很简单,线性方程组不需要任何额外的自由度。本制剂在WGM中治疗BCs的准确性很高,这通过包括适应性分析在内的许多代表性数值实例得以说明。

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