Verlag der Österreichischen Akademie der Wissenschaften

Abstract:
In this paper we analyze the numerical approximation of a saddle-point problem posed innon-standard Banach spaces $\mathrm{H}(\mathrm{div}_{p}\,, \Omega)\times L^q(\Omega)$, where$\mathrm{H}(\mathrm{div}_{p}\,, \Omega):= \{{\boldsymbol\tau} \in [L^2(\Omega)]^n \colon \mathrm{div} {\boldsymbol\tau} \in L^p(\Omega)\},$with $p>1$ and $q\in \mathbb{R}$ being the conjugate exponent of $p$ and $\Omega\subseteq \mathbb{R}^n$ ($n\in\{2,3\}$)a bounded domain with Lipschitz boundary $\Gamma$. In particular, we are interestedin deriving the stability properties of the forms involved (inf-sup conditions, boundedness),which are the main ingredients to analyze mixed formulations. In fact, by using these propertieswe prove the well-posedness of the corresponding continuous and discrete saddle-point problems by means of theclassical Babuška-Brezzi theory, where the associated Galerkin scheme is defined by Raviart-Thomaselements of order $k\geq 0$ combined with piecewise polynomials of degree $k$. In addition weprove optimal convergence of the numerical approximation in the associated Lebesgue norms.Next, by employing the theory developed for the saddle-point problem, weanalyze a mixed finite element method for a convection-diffusion problem, providing well-posedness of the continuousand discrete problems and optimal convergence under a smallness assumption on the convective vector field.Finally, we corroborate the theoretical results with suitable numerical results in two and three dimensions.