On Numerical Methods for Hyperbolic Conservation Laws and Related Equations Modelling Sedimentation of Solid-Liquid Suspensions

Concha, F.; Bürger, Raimund; Ruiz–Baier, Ricardo; Torres, Héctor; Vega, Carlos A.

doi:10.1007/978-3-642-39007-4_2

Cited by 1 publication

(1 citation statement)

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“…This extension (whose limit case translates into a loss of ellipticity in the concentration equation) implies that for low volume fractions, one may expect shock-like fronts to develop (see e.g. the monograph [9] and the recent review [5]). Adaptive mesh refinement would then be highly appreciated in this particular case; not only to restitute optimal convergence orders, but also to resolve concentration profiles accurately without the need of refining the grid everywhere (even more important if higher-order schemes are used, or transient models are studied).…”

Section: Numerical Testsmentioning

confidence: 99%

A posteriorierror analysis for a viscous flow-transport problem

Alvarez¹,

Gatica²,

Ruiz–Baier³

2016

ESAIM: M2AN

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In this paper we develop an a posteriori error analysis for an augmented mixed-primal finite element approximation of a stationary viscous flow and transport problem. The governing system corresponds to a scalar, nonlinear convection-diffusion equation coupled with a Stokes problem with variable viscosity, and it serves as a prototype model for sedimentation-consolidation processes and other phenomena where the transport of species concentration within a viscous fluid is of interest. The solvability of the continuous mixed-primal formulation along with a priori error estimates for a finite element scheme using Raviart-Thomas spaces of order k for the stress approximation, and continuous piecewise polynomials of degree ≤ k + 1 for both velocity and concentration, have been recently established in [M. Alvarez et al., ESAIM: Math. Model. Numer. Anal. 49 (5) (2015) 1399-1427]. Here we derive two efficient and reliable residual-based a posteriori error estimators for that scheme: For the first estimator, and under suitable assumptions on the domain, we apply a Helmholtz decomposition and exploit local approximation properties of the Clément interpolant and Raviart-Thomas operator to show its reliability. On the other hand, its efficiency follows from inverse inequalities and the localization arguments based on triangle-bubble and edge-bubble functions. Secondly, an alternative error estimator is proposed, whose reliability can be proved without resorting to Helmholtz decompositions. Our theoretical results are then illustrated via some numerical examples, highlighting also the performance of the scheme and properties of the proposed error indicators.

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Section: Numerical Testsmentioning

confidence: 99%