We propose and analyse new space-time Galerkin-Bubnov-type finite element formulations of parabolic and hyperbolic second-order partial differential equations in finite time intervals. Using Hilbert-type transformations, this approach is based on elliptic reformulations of first-and second-order time derivatives, for which the Galerkin finite element discretisation results in positive definite and symmetric matrices. For the variational formulation of the heat and wave equations, we prove related stability conditions in appropriate norms, and we discuss the stability of related finite element discretisations. Numerical results are given which confirm the theoretical results.

AbstractIn this note we consider an efficient data–sparse approximation of a modified Hilbert type transformation as it is used for the space–time finite element discretization of parabolic evolution equations in the anisotropic Sobolev space H1,1/2(Q). The resulting bilinear form of the first order time derivative is symmetric and positive definite, and similar as the integration by parts formula for the Laplace hypersingular boundary integral operator in 2D. Hence we can apply hierarchical matrices for data–sparse representations and for acceleration of the computations. Numerical results show the efficiency in the approximation of the first order time derivative. An efficient realisation of the modified Hilbert transformation is a basic ingredient when considering general space–time finite element methods for parabolic evolution equations, and for the stable coupling of finite and boundary element methods in anisotropic Sobolev trace spaces.

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.

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