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.
Highlights
For the discretization of parabolic evolution equations, the classical approaches are time stepping schemes together with finite element methods in space
There are space-time discretizations of parabolic evolution equations based on the variational formulations in Bochner–Sobolev spaces, see, e.g., [1, 5, 8, 10, 11, 16,17,18, 21, 24]
We propose a new series representation of the modified Hilbert transformation HT for piecewise polynomial functions, which converges very fast independently of the time mesh size ht,min, i.e. only a few terms in this new series are needed to calculate the matrix entries of
Summary
For the discretization of parabolic evolution equations, the classical approaches are time stepping schemes together with finite element methods in space. Discretizations of variational formulations in anisotropic Sobolev spaces. We consider the space-time variational formulation of (1.1) to find u ∈ H01;,10/,2∙ (Q) such that a(u, v) = ⟨f, v⟩Q (1.2). In the remainder of this work, we consider p = 1, i.e. the tensor-product space of piecewise multilinear, continuous functions Vh = Q1h(Q) ∩ H01;,10/,2∙ (Q), where analogous results hold true for an arbitrary polynomial degree p > 1. We propose a new series representation of the modified Hilbert transformation HT for piecewise polynomial functions, which converges very fast independently of the time mesh size ht,min, i.e. only a few terms in this new series are needed to calculate the matrix entries of HT ht.
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