We present different possibilities of realizing a modified Hilbert type transformation as it is used for Galerkin–Bubnov discretizations of space-time variational formulations for parabolic evolution equations in anisotropic Sobolev spaces of spatial order 1 and temporal order 1 2 \frac{1}{2} . First, we investigate the series expansion of the definition of the modified Hilbert transformation, where the truncation parameter has to be adapted to the mesh size. Second, we introduce a new series expansion based on the Legendre chi function to calculate the corresponding matrices for piecewise polynomial functions. With this new procedure, the matrix entries for a space-time finite element method for parabolic evolution equations are computable to machine precision independently of the mesh size. Numerical results conclude this work.

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抛物线发展方程的时空方法的修正希尔伯特变换的精确实现

我们提出了一种修正的希尔伯特类型变换的不同可能性，因为该变换用于空间序1和时间序1 2的各向异性Sobolev空间中抛物线发展方程的时空变式的Galerkin–Bubnov离散化2 \ frac {1} {2 }。首先，我们研究了修改后的希尔伯特变换的定义的级数展开，其中截断参数必须适应网格大小。其次，我们引入了一个基于Legendre chi函数的新的级数展开，以计算分段多项式函数的相应矩阵。通过这种新程序，抛物线发展方程的时空有限元方法的矩阵项可独立于网格大小而计算出机器精度。数值结果总结了这项工作。