Error assessment for functional outputs of pde's: bounds and goal-oriented adaptivity

• Autores: Núria Parés Marine
• Directores de la Tesis: Antonio Huerta Cerezuela (dir. tes.) , Pedro Díez Mejía (dir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2005
• Idioma: español
• Tribunal Calificador de la Tesis: Jaume Peraire Guitar (presid.) , Irene Arias Vicente (secret.) , Guillermo Hauke Bernardos (voc.) , Paulo Moitihno Jose (voc.) , Pierre Ladeveze (voc.)
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• Resumen

  • Engineering applications typically consist in studying a physical phenomena in order to predict certain quantities relevant to the analysis or decision-making process such as averages of the solution, flow rates, velocities or shear stress at a given critical point in the domain and heat transfer, It is frequent to study if a design meets the security requirements or to study how to modify a design in order to improve its performance requirements, which are the quantities of analysis. To approximate this quantities numerical approximations of the physical phenomena are used. Therefore, the accuracy of the numerical results is given by its capacity to provide reliable quantitative information about the quantities of interest also called outputs. The obtention of an approximated solution with a global prescribed accuracy is not the main goal but rather the control of the error in the output, which represents the relevant engineering quantity.

    This thesis is focused in the verification of numerical results, that is, in the evaluation of the errors introduced in the discretization process of transforming the mathematical model problem into a numerical problem.

    The goal is to control and assess the discretization error. In particular, special interest is paid in the assessment of the discretization error not only in a global norm but in a particular quantity of interest.

    First of all a general framework to obtain bounds for outputs of interest is presented relating the obtention of bounds for outputs of interest with the obtention of upper and lower bounds for the energy norm both for symmetric an non-symmetric coercive model problems. Bounds for the output may be computed from upper bounds for the energy norm and the obtained bounds may be enhanced if also lower bounds for the error are available. This motivates the first contribution of the thesis, the obtention of lower bounds of the energy norm by post-processing implicit residual a po