A posteriori error analysis for a continuous space-time finite element method for a hyperbolic integro-differential equation

Continuous Galerkin finite element methods for hyperbolic integro-differential equations

“…This is then used to prove L ∞ (L 2 ) and L ∞ (H 1 ) optimal order a priori error estimates by duality. This and [14], where a posteriori error analysis of this method has been studied via duality, complete the error analysis of this method for model problems similar to (1.1).…”

“…This construction is necessary to allow for trial functions that are continuous also at the discrete time leveles even if grids change between time steps. For more details and computational aspects, including hanging nodes, see [14] and the references therein. Associating triangulation with time slabs instead of time levels would yield a variant scheme which includes jump terms due to discontinuity at discrete time leveles, when coarsening happens.…”

“…and, having (1.2), it is easy to see that D t ξ(t) = −K(t) < 0, ξ(0) = κ, lim t→∞ ξ(t) = 0, 0 < ξ(t) ≤ κ. (1.4) Hence, ξ is a completely monotone function, since (−1) j D j t ξ(t) ≥ 0, t ∈ (0, ∞), j = 0, 1, 2, and consequently ξ ∈ L 1,loc [0, ∞) is a positive type kernel, that is, for any T ≥ 0 and φ ∈ C([0, T ]), From the extensive literature on theoritical and numerical analysis for partial differential equations with memory, we mention [13], [7], [2], [10], [14], and their references.…”