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

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 \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.

Towards coercive boundary element methods for the wave equation

“…Here, for an easier implementation, we approximate the right-hand side g ∈ H 1/2 0, (Σ) by Q h g, where Q h is the L 2 projection on the space of piecewise linear, continuous functions fulfilling homogeneous initial conditions for t = 0. The assembling of the matrix V h ∈ R (N 0 +N L )×(N 0 +N L ) and the right-hand side g ∈ R N 0 +N L , i.e., the realization of H T , is done as proposed in [19,Subsection 2.2]. The integrals for computing the projection Q h g are calculated by using high-order quadrature rules.…”

“…0, (Σ) in combination with a modified Hilbert transformation [15,16,19] even satisfies an ellipticity estimate similar as in (2.7).…”

Adaptive space‐time finite element solvers for parabolic initial‐boundary value problems with non‐smooth solutions

“…The novel transformation H T acts on the finite terminal (0, T ), whereas analogous considerations of an infinite time interval (0, ∞) with the classical Hilbert transformation are investigated in [8,11,12,24]. The most important properties of H T are summarized in the following, see [41,42,47,48]. The map…”

Reliable Methods of Mathematical Modeling

“…The work [12] of Zank considers a space-time discretization of parabolic evolution equations in the setting of anisotropic space-time Sobolev spaces. While classical discretizations of evolution problems rely on time marching schemes, the beneőts of space-time methods are that they have the potential for space-time adaptivity as well as parallelization.…”

“…This special issue collects selected works from participants of RMMM 2019 that are related to their presentations. The overall focus is as wide as the needs for mathematics in computational PDEs, addressing a posteriori error control [2,5] and adaptivity [1,3,7,8], reliable methods for space-time problems [3,4,12], non-standard numerical discretizations [2,4,6,9], and iterative solvers and optimal preconditioning [7,8,10,11].…”