Coercive space-time finite element methods for initial boundary value problems

Abstract

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.

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

“…see [25,26,41,48] for more details. The dual space [H 1,1/2 0; ,0 (Q)] is characterized as completion of L 2 (Q) with respect to the Hilbertian norm…”

“…x } and with a constant c > 0 for a sufficiently smooth solution u ∈ H 1,1/2 0;0, (Q) of (3) and a sufficiently regular boundary ∂Ω, where for the H 1 (Q) error estimate (13), the sequence (T ν ) ν∈N of decompositions of Ω is additionally assumed to be globally quasi-uniform, see [41,48] for details.…”

“…In this work, the approach in anisotropic Sobolev spaces is applied in combination with a novel Hilbert-type transformation operator H T , which has recently been introduced in [41,48]. This transformation operator H T maps the ansatz space to the test space, and gives a symmetric and elliptic variational setting of the first-order time derivative.…”

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

“…On the one hand, 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, 16-18, 21, 24]. On the other hand, discretizations of variational formulations in anisotropic Sobolev spaces of spatial order 1 and temporal order 1 2 became quite attractive recently, see, e.g., [4,12,19,23,25]. In this work, the approach in these anisotropic Sobolev spaces is applied.…”

“…where Ω ⊂ ℝ d , d = 1, 2, 3, is a bounded Lipschitz domain with boundary ∂Ω, T > 0 is a given terminal time and f is a given right-hand side. Next, we consider the space-time variational formulation of (1.1) to find u ∈ H • ,0 (0, T; L 2 (Ω)) := {w ∈ H 1/2 (0, T; L 2 (Ω)) : ‖w‖ H 1/2 • ,0 (0,T;L 2 (Ω)) < ∞}, see [14,15,23,25] for more details. Moreover, the dual space [H…”