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In this work a p-adaptation (modification of the polynomial order) strategy based on the minimization of the truncation error is developed for high order discontinuous Galerkin methods. The truncation error is approximated by means of a truncation error estimation procedure and enables the identification of mesh regions that require adaptation. Three truncation error estimation approaches are developed and termed a posteriori, quasi-a priori and quasi-a priori corrected. Fine solutions, which are obtained by enriching the polynomial order, are required to solve the numerical problem with adequate accuracy. For the three truncation error estimation methods the former needs time converged solutions, while the last two rely on non-converged solutions, which lead to faster computations. Based on these truncation error estimation methods, algorithms for mesh adaptation were designed and tested. Firstly, an isotropic adaptation approach is presented, which leads to equally distributed polynomial orders in different coordinate directions. This first implementation is improved by incorporating a method to extrapolate the truncation error. This results in a significant reduction of computational cost. Secondly, the employed high order method permits the spatial decoupling of the estimated errors and enables anisotropic p-adaptation. The incorporation of anisotropic features leads to meshes with different polynomial orders in the different coordinate directions such that flow-features related to the geometry are resolved in a better manner. These adaptations result in a significant reduction of degrees of freedom and computational cost, while the amount of improvement depends on the test-case. Finally, this anisotropic approach is extended by using error extrapolation which leads to an even higher reduction in computational cost. These strategies are verified and compared in terms of accuracy and computational cost for the Euler and the compressible Navier-Stokes equations. The main result is that the two quasi-a priori methods achieve a significant reduction in computational cost when compared to a uniform polynomial enrichment. Namely, for a viscous boundary layer flow, we obtain a speedup of a factor of 6.6 and 7.6 for the quasi-a priori and quasi-a priori corrected approaches, respectively. RESUMEN En este trabajo se ha desarrollado una estrategia de adaptacion-p (modificacion del orden polinomico) para metodos Galerkin discontinuo de alto orden basada en la minimizacion del error de truncacion. El error de truncacion se estima utilizando el metodo tau-estimation. El estimador permite la identificacion de zonas de la malla que requieren adaptacion. Se distinguen tres tecnicas de estimacion: a posteriori, quasi a priori y quasi a priori con correcion. Todas las estrategias requieren una solucion obtenida en una malla fina, la cual es obtenida aumentando de manera uniforme el orden polinomico. Sin embargo, mientras que el primero requiere que esta solucion este convergida temporalmente, el resto utiliza soluciones no convergidas, lo que se traduce en un menor coste computacional. En este trabajo se han disenado y probado algoritmos de adaptacion de malla basados en metodos tau-estimation. En primer lugar, se presenta un algoritmo de adaptacin isotropo, que conduce a discretizaciones con el mismo orden polinomico en todas las direcciones espaciales. Esta primera implementacion se mejora incluyendo un metodo para extrapolar el error de truncacion. Esto resulta en una reduccion significativa del coste computacional. En segundo lugar, el metodo de alto orden permite el desacoplamiento espacial de los errores estimados, permitiendo la adaptacion anisotropica. Las mallas obtenidas mediante esta tecnica tienen distintos ordenes polinomicos en cada una de las direcciones espaciales. La malla final tiene una distribucion optima de ordenes polinomicos, los cuales guardan relacion con las caracteristicas del flujo que, a su vez, depenen de la geometria. Estas tecnicas de adaptacion reducen de manera significativa los grados de libertad y el coste computacional. Por ultimo, esta aproximacion anisotropica se extiende usando extrapolacion del error de truncacion, lo que conlleva un coste computational aun menor. Las estrategias se verifican y se comparan en teminors de precision y coste computacional utilizando las ecuaciones de Euler y Navier Stokes. Los dos metodos quasi a priori consiguen una reduccion significativa del coste computacional en comparacion con aumento uniforme del orden polinomico. En concreto, para una capa limite viscosa, obtenemos una mejora en tiempo de computacion de 6.6 y 7.6 respectivamente, para las aproximaciones quasi-a priori y quasi-a priori con correccion.

Adaptation Strategies for Discontinuous Galerkin Spectral Element Methods by means of Truncation Error Estimations