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.
翻译：我们提出了计算矩阵的二次曲线方案,其中涉及所谓的修改的希尔伯特转换。这些矩阵是作为Galerkin 有限元素分解的时段部分出现的。当修改的希尔伯特变异用于变异设置时,这些矩阵是抛物线或双曲问题的抛物体或双曲问题。这项工作将这些矩阵计算成机器精度的机器精确度,以任意多角度和非单形色体。拟议的二次曲线方案是以修改的希尔伯特变异的微弱单一整体表示法的单一整体表示法为基础。首先,这些经修改的希尔伯特变异的微弱单一整体表示法得到了证明。第二,利用这些整体表示法,我们产生了对正在发生的奇特性进行适当处理的二次曲线方案。因此,在拟议的二次曲线组合计划中的二次曲线节点数量方面实现了指数趋同。在观察到这种指数趋同的情况下,数字结果将结束这项工作。
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