Numerical analysis of a dual-mixed problem in non-standard Banach spaces

Jessika Camaño, Cristian Muñoz, and Ricardo Oyarzúa

Abstract

In this paper we analyze the numerical approximation of a saddle-point problem posed in non-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 interested in 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 properties we prove the well-posedness of the corresponding continuous and discrete saddle-point problems by means of the classical Babuška-Brezzi theory, where the associated Galerkin scheme is defined by Raviart-Thomas elements of order $k\geq 0$ combined with piecewise polynomials of degree $k$. In addition we prove optimal convergence of the numerical approximation in the associated Lebesgue norms. Next, by employing the theory developed for the saddle-point problem, we analyze a mixed finite element method for a convection-diffusion problem, providing well-posedness of the continuous and 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.

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Key words

mixed finite element method, Raviart-Thomas, Lebesgue spaces, Lp data, convection-diffusion

AMS subject classifications

65N15, 65N12, 65N30, 74S05

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