Variational multiscale a posteriori error estimation in finite element methods for fluid mechanics and elasticity
- Autores: Diego Irisarri Jiménez
- Directores de la Tesis: Guillermo Hauke Bernardos (dir. tes.)
- Lectura: En la Universidad de Zaragoza ( España ) en 2017
- Idioma: español
- Tribunal Calificador de la Tesis: Pierre Ladeveze (presid.), Antonio Huerta Cerezuela (secret.), Doweidar Taky el Din Mohamed Hamdy Doweidar (voc.)
- Materias:
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- Resumen
The present thesis is concerned with a posteriori error estimation for several differential equations of elasticity and fluid mechanics solved by the finite element method.
The error estimators are developed in the framework of the variational multiscale theory [1,2], in which the exact solution is split into coarse (or resolved) and fine (or unresolved) scales. In combination with fine-scale Green's functions [3], residual-free bubbles and classic Green's functions, the fine scales are described and their interaction with the coarse scales is established. The error estimation has been accomplished following previous works such as [4,5,6]. According to the nature of the residuals, the error is decomposed in two components: the internal residual error and the inter-element error. In particular, the internal residual error, which has a local character, is modeled with residual-free bubbles or a combination of bubble functions, whereas the inter-element error, which presents a global character, is constituted by classic Green's functions.
Using the above model, new explicit and implicit error estimators have been developed. In the literature, we can found a wide number of error estimators both in elasticity and fluid mechanics [7,8,9,10,11]. In this work, global, local and pointwise error estimates have been proposed. The local and global error estimates are constant-free, economical and simple to implement in existing codes. Furthermore, the model yields an accurate pointwise error representation, providing exact pointwise error estimates in one-dimensional problems regardless of the element order. Numerical examples confirm the theoretical framework of this methodology. Adaptive mesh refinement has been carried out allowing to assess the pointwise or elemental error.
The error estimation is applied to elasticity for one- and two-dimensional problems [12,13]. For 1D, second and fourth ODEs are considered. In 2D problems, we deal with plane stress problems. As for fluid mechanics, error estimators are proposed for the transport equation and the Stokes equations. Particularly, for the transport equation, the reaction-diffusion and the convection-diffusion equations are considered [14,15]. For the Stokes equations, the error estimation has been carried out for stabilized methods. Implicit and explicit error estimators are presented.
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[15] Irisarri, D., Hauke, G.: A posteriori pointwise error computation for 2-D transport equations based on the variational multiscale method. Computer Methods in Applied Mechanics and Engineering 311, 648–670 (2016)
