On the continuous and discrete well-posedness of perturbed saddle-point formulations in Banach spaces

Abstract

In this paper we provide sufficient conditions for perturbed saddle-point formulations in Banach spaces and their associated Galerkin schemes to be well-posed. Our approach, which extends a similar procedure employed with Hilbert spaces, proceeds in two slightly different ways depending on whether the kernel of the adjoint operator induced by one of the bilinear forms is trivial or not. If the latter holds, we make use of an equivalence result between a couple of inf-sup conditions involved, which, differently from the Hilbertian case, turns out to hold with different constants. While this fact causes no inconvenient in the continuous analysis, it does become a delicate issue at the discrete level, and hence the corresponding inf-sup conditions need to be assumed separately with respective constants independent of the finite element subspaces employed. In turn, if that kernel is trivial, then we employ a suitable characterization of a closed range injective adjoint operator, so that only one of the aforementioned inf-sup conditions is then required for the analysis. The applicability of the continuous solvability is illustrated with a mixed formulation for the decoupled Nernst-Planck equation.