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

Abstract

Abstract. In this paper we analyze the numerical approximation of a saddle-point problem posed in non-standard Banach spaces H(divp , Ω) × Lq (Ω), where H(divp , Ω) := {τ ∈ [L2 (Ω)]n : divτ ∈ Lp(Ω)}, with p > 1 and q ∈ R being the conjugate exponent of p and Ω ⊆ Rn (n ∈ {2, 3}) a bounded domain with Lipschitz boundary Γ. 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 ≥ 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.