113 research outputs found
Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids
In this paper, we first construct a nonconforming finite element pair for the
incompressible Stokes problem on quadrilateral grids, and then construct a
discrete Stokes complex associated with that finite element pair. The finite
element spaces involved consist of piecewise polynomials only, and the
divergence-free condition is imposed in a primal formulation. Combined with
some existing results, these constructions can be generated onto grids that
consist of both triangular and quadrilateral cells
Superconvergence of a nonconforming brick element for the quad-curl problem
This short note shows the superconvergence of an
-nonconforming brick element very recently
introduced in [17] for the quad-curl problem. The supercloseness is based on
proper modifications for both the interpolation and the discrete formulation,
leading to an superclose order in the discrete
norm. Moreover, we propose a suitable
postprocessing method to ensure the global superconvergence. Numerical results
verify our theory
Stable cheapest nonconforming finite elements for the Stokes equations
We introduce two pairs of stable cheapest nonconforming finite element space
pairs to approximate the Stokes equations. One pair has each component of its
velocity field to be approximated by the nonconforming quadrilateral
element while the pressure field is approximated by the piecewise constant
function with globally two-dimensional subspaces removed: one removed space is
due to the integral mean--zero property and the other space consists of global
checker--board patterns. The other pair consists of the velocity space as the
nonconforming quadrilateral element enriched by a globally
one--dimensional macro bubble function space based on
(Douglas-Santos-Sheen-Ye) nonconforming finite element space; the pressure
field is approximated by the piecewise constant function with mean--zero space
eliminated. We show that two element pairs satisfy the discrete inf-sup
condition uniformly. And we investigate the relationship between them. Several
numerical examples are shown to confirm the efficiency and reliability of the
proposed methods
Adaptive low and high-order hybridized methods for unsteady incompressible flow simulations
Tesi en modalitat de cotutela: Universitat Politècnica de Catalunya i Università degli Studi di PaviaSimulations of incompressible flows are performed on a daily basis to solve problems of practical and industrial interest in several fields of engineering, including automotive, aeronautical, mechanical and biomedical applications.
Although finite volume (FV) methods are still the preferred choice by the industry due to their efficiency and robustness, sensitivity to mesh quality and limited accuracy represent two main bottlenecks of these approaches. This is especially critical in the context of transient phenomena, in which FV methods show excessive numerical diffusion. In this context, there has been a growing interest towards high-order discretisation strategies in last decades.
In this PhD thesis, a high-order adaptive hybidisable discontinuous Galerkin (HDG) method is proposed for the approximation of steady and unsteady laminar incompressible Navier-Stokes equations. Voigt notation for symmetric second-order tensors is exploited to devise an HDG method for the Cauchy formulation of the momentum equation with optimal convergence properties, even when low-order polynomial degrees of approximation are considered. In addition, a postprocessing strategy accounting for rigid translational and rotational modes is proposed to construct an element-by-element superconvergent velocity field.
The discrepancy between the computed and postprocessed velocities is utilised to define a local error indicator to drive degree adaptivity procedures and accurately capture localised features of the flow. The resulting HDG solver is thus extended to the case of transient problems via high-order time integration schemes, namely the explicit singly diagonal implicit Runge-Kutta (ESDIRK) schemes. In this context, the embedded explicit step is exploited to define an inexpensive estimate of the temporal error to devise an efficient timestep control strategy.
Finally, in order to efficiently solve the global problem arising from the HDG discretisation, a preconditioned iterative solver is proposed. This is critical in the context of high-order approximations in three-dimensional domains leading to large-scale problems, especially in transient simulations. A block diagonal preconditioner coupled with an inexpensive approximation of the Schur complement of the matrix is proposed to reduce the computational cost of the overall HDG solver.
Extensive numerical validation of two and three-dimensional steady and unsteady benchmark tests of viscous laminar incompressible flows is performed to validate the proposed methodology.Simulaciones de flujo incompresible se emplean a diario para resolver problemas de interés práctico e industrial en varios campos de la ingenierÃa, p.ej. en aplicaciones automovilÃsticas, aeronáuticas, mecánicas y biomédicas. Aunque los métodos de volúmenes finitos (FV) siguen siendo la opción preferida por la industria debido a su eficiencia y robustez, la sensibilidad a la calidad de la malla y la baja precisión representan dos limitaciones importantes para estas técnicas. Estas limitaciones son todavÃa más crÃticas en el contexto de simulaciones de fenómenos transitorios, donde los FV están penalizados por su excesiva difusión numérica. En este contexto, las estrategias de discretización de alto orden han ganado una popularidad creciente en las últimas décadas para problemas transitorios dónde se necesitan soluciones precisas. Esta tesis propone un método de Galerkin discontinuo hÃbrido (HDG), de alto orden y adaptativo para la aproximación de las ecuaciones de Navier-Stokes incomprensible laminar, en el caso estacionario y transitorio en el entorno de aplicaciones ingenieriles. Para ello, la notación de Voigt para tensores simétricos de segundo orden (habituales en mecánica de los medios continuos) permite introducir un método HDG para la formulación de Cauchy de la ecuación de momento. La novedad de este resultado reside en la convergencia óptima alcanzada por el método, incluso para aproximaciones de orden polinómico bajo. Además, se desarrolla una estrategia de post-proceso local para construir elemento a elemento un campo de velocidad súper-convergente, tomando en cuenta los modos rÃgidos de traslación y rotación. La discrepancia entre el campo de velocidad calculado y el súper-convergente, obtenido a través del post-proceso, permite definir un indicador del error local. De esta forma, se desarrolla una estrategia para realizar adecuar elemento a elemento el grado de la aproximación polinómica y asà mejorar la precisión adaptándose a las caracterÃsticas localizadas del flujo. Seguidamente, se extiende el método HDG propuesto al tratamiento de problemas dependientes del tiempo. Más concretamente, se consideran los esquemas de integración temporal de alto orden explicit singly diagonal implicit Runge-Kutta (ESDIRK). En este contexto, se utiliza el paso explÃcito embedded para calcular una estimación computacionalmente eficiente del error temporal y definir una estrategia de adaptividad del paso de tiempo. Finalmente, se desarrolla un precondicionador adaptado a la estrategia HDG que acelera la convergencia del método iterativo empleado y, de esta forma, obtener resoluciones eficaces del problema global surgido de la discretización HDG. Es importante resaltar la importancia de una herramienta de resolución eficiente para problemas de gran escala en el contexto de aproximaciones de alto orden y en dominios tridimensionales. Estas herramientas se hacen aún más criticas en simulaciones transitorias. Más concretamente, se proponen un precondicionador diagonal por bloques y una aproximación eficiente del complemento Schur de la matriz para reducir el coste computacional del método HDG. Para validar la metodologÃa propuesta, se realizan varias simulaciones numéricas de flujo incompresible laminar viscoso, para problemas estacionarios y transitorios, en dos y tres dimensiones.Postprint (published version
Analysis of a finite volume element method for the Stokes problem
In this paper we propose a stabilized conforming finite volume element method for the Stokes equations. On stating the convergence of the method, optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations. A superconvergence result is also derived by using a postprocessing projection method. In particular, the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries. Finally, some numerical experiments that confirm the predicted behavior of the method are provide
- …