论文标题
修改后的希尔伯特转换的积分表示和正交方案
Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation
论文作者
论文摘要
我们提出了正交方案来计算矩阵，其中涉及所谓的修改后的希尔伯特转换。这些矩阵作为抛物线或双曲线问题的Galerkin有限元离散的时间部分出现，当修改后的希尔伯特转换用于变异设置时。这项工作为任意多项式度和不均匀网格的机器精度提供了这些矩阵的计算。提出的正交方案基于修改后的希尔伯特转化的弱奇异积分表示。首先，证明了修改后的希尔伯特转型的这些弱奇异的积分表示。其次，使用这些积分表示，我们得出了正交方案，这些方案适当地处理发生的奇异性。因此，实现了针对提出的正交方案的正交节点数量的指数收敛。观察到这一指数收敛的数值结果得出了这项工作。
We present quadrature schemes to calculate matrices, where the so-called modified Hilbert transformation is involved. These matrices occur as temporal parts of Galerkin finite element discretizations of parabolic or hyperbolic problems when the modified Hilbert transformation is used for the variational setting. This work provides the calculation of these matrices to machine precision for arbitrary polynomial degrees and non-uniform meshes. The proposed quadrature schemes are based on weakly singular integral representations of the modified Hilbert transformation. First, these weakly singular integral representations of the modified Hilbert transformation are proven. Second, using these integral representations, we derive quadrature schemes, which treat the occurring singularities appropriately. Thus, exponential convergence with respect to the number of quadrature nodes for the proposed quadrature schemes is achieved. Numerical results, where this exponential convergence is observed, conclude this work.
