An Exact Realization of a Modified Hilbert Transformation for Space-Time Methods for Parabolic Evolution Equations

Author:

Zank Marco1ORCID

Affiliation:

1. Faculty of Mathematics , University of Vienna , Oskar-Morgenstern-Platz 1, A-1090 Wien , Austria

Abstract

Abstract 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.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Computational Mathematics,Numerical Analysis

Reference25 articles.

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