arXiv subjects

P. A. Gazca-Orozco

Publications and source records attributed to P. A. Gazca-Orozco.

7 recordsLinked to original sources

arXiv2026

Pressure-robust and quasioptimal Discontinuous Galerkin discretisations of the $p$-Stokes problem

In the present paper, we propose Local Discontinuous Galerkin (LDG) approximations for a nonlinear system of $p$-Stokes type, having $(p,δ)$-structure. On the basis of the primal formulation, we prove well-posedness and stability (a priori estimates) of the methods under truly minimal regularity assumptions. We show that the first method possesses a pressure-robust and quasi-optimal error estimate, and discuss its consequences. Moreover, we propose a second method, for which we show a pressure-robust error estimate and prove convergence and convergence rates, which are optimal for linear ansatz functions for all $p\in (1,\infty)$ and $δ\geq 0$.

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arXiv2026

Bregman divergences and error control via convex duality

Convex duality relations are a useful tool for deriving error estimates for challenging nonlinear and non-smooth variational problems. Applied at the continuous level they can deliver nonlinear analogues of the Prager-Synge a posteriori error identity, while at the discrete level they allow the derivation of minimal regularity a priori estimates. By leveraging elementary properties of Bregman divergences, we obtain three results on the error control via convex duality for a general class of problems: first, we prove a local efficiency bound for the duality gap error estimator, secondly, we derive a guaranteed a posteriori bound for non-conforming fields, and finally, we prove a minimal-regularity quasioptimal estimate for a Crouzeix-Raviart discretisation of the $φ$-Laplace problem.

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arXiv2026

A Priori and A Posteriori Error Identities for Vectorial Problems via Convex Duality

Convex duality has been leveraged in recent years to derive a posteriori error estimates and identities for a wide range of non-linear and non-smooth scalar problems. By employing remarkable compatibility properties of the Crouzeix-Raviart and Raviart-Thomas elements, optimal convergence of non-conforming discretisations and flux reconstruction formulas have also been established. This paper aims to extend these results to the vectorial setting, focusing on the archetypical problems of incompressible Stokes and Navier-Lamé. Moreover, unlike most previous results, we consider inhomogeneous mixed boundary conditions and loads in the topological dual of the energy space. At the discrete level, we derive error identities and estimates that enable to prove quasi-optimal error estimates for a Crouzeix-Raviart discretisation with minimal regularity assumptions and no data oscillation terms.

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arXiv2023

Quasi-optimal Discontinuous Galerkin discretisations of the $p$-Dirichlet problem

The classical arguments employed when obtaining error estimates of Finite Element (FE) discretisations of elliptic problems lead to more restrictive assumptions on the regularity of the exact solution when applied to non-conforming methods. The so-called minimal regularity estimates available in the literature relax some of these assumptions, but are not truly of -minimal regularity-, since a data oscillation term appears in the error estimate. Employing an approach based on a smoothing operator, we derive for the first time error estimates for Discontinuous Galerkin (DG) type discretisations of non-linear problems with $(p,δ)$-structure that only assume the natural $W^{1,p}$-regularity of the exact solution, and which do not contain any oscillation terms.

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arXiv2023

Development of boundary layers in Euler fluids that on "activation'' respond like Navier-Stokes fluids

We consider the flow of a fluid whose response characteristics change due the value of the norm of the symmetric part of the velocity gradient, behaving as an Euler fluid below a critical value and as a Navier-Stokes fluid at and above the critical value, the norm being determined by the external stimuli. We show that such a fluid, while flowing past a bluff body, develops boundary layers which are practically identical to those that one encounters within the context of the classical boundary layer theory propounded by Prandtl. Unlike the classical boundary layer theory that arises as an approximation within the context of the Navier-Stokes theory, here the development of boundary layers is due to a change in the response characteristics of the constitutive relation. We study the flow of such a fluid past an airfoil and compare the same against the solution of the Navier-Stokes equations. We find that the results are in excellent agreement with regard to the velocity and vorticity fields for the two cases.

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arXiv2021

A semismooth Newton method for implicitly constituted non-Newtonian fluids and its application to the numerical approximation of Bingham flow

We propose a semismooth Newton method for non-Newtonian models of incompressible flow where the constitutive relation between the shear stress and the symmetric velocity gradient is given implicitly; this class of constitutive relations captures for instance the models of Bingham and Herschel-Bulkley. The proposed method avoids the use of variational inequalities and is based on a particularly simple regularisation for which the (weak) convergence of the approximate stresses is known to hold. The system is analysed at the function space level and results in mesh-independent behaviour of the nonlinear iterations.

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arXiv2020

An augmented Lagrangian preconditioner for implicitly-constituted non-Newtonian incompressible flow

We propose an augmented Lagrangian preconditioner for a three-field stress-velocity-pressure discretization of stationary non-Newtonian incompressible flow with an implicit constitutive relation of power-law type. The discretization employed makes use of the divergence-free Scott-Vogelius pair for the velocity and pressure. The preconditioner builds on the work [P. E. Farrell, L. Mitchell, and F. Wechsung, SIAM J. Sci. Comput., 41 (2019), pp. A3073-A3096], where a Reynolds-robust preconditioner for the three-dimensional Newtonian system was introduced. The preconditioner employs a specialized multigrid method for the stress-velocity block that involves a divergence-capturing space decomposition and a custom prolongation operator. The solver exhibits excellent robustness with respect to the parameters arising in the constitutive relation, allowing for the simulation of a wide range of materials.

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