Volume 48, pp. 114-130, 2018.

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 $\mathrm{H}({\mathrm{div}}_p,\mathrm{\Omega })\times L^q(\mathrm{\Omega })$, where $\mathrm{H}({\mathrm{div}}_p,\mathrm{\Omega }):=\{\mathit{\tau }\in [L^2(\mathrm{\Omega }){]}^n:\mathrm{div}\mathit{\tau }\in L^p(\mathrm{\Omega })\},$ with $p>1$ and $q\in \mathbb{R}$ being the conjugate exponent of $p$ and $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ ($n\in \{2,3\}$) a bounded domain with Lipschitz boundary $\mathrm{\Gamma }$. 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 $k\ge 0$ 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.

Full Text (PDF) [742 KB], BibTeX , DOI: 10.1553/etna_vol48s114

Key words

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

AMS subject classifications

65N15, 65N12, 65N30, 74S05