h1

%0 Thesis
%A Chakraborty, Ankit
%T Optimal approximation spaces for discontinuous Petrov-Galerkin schemes with optimal test functions
%I Rheinisch-Westfälische Technische Hochschule Aachen
%V Dissertation
%C Aachen
%M RWTH-2022-08228
%P 1 Online-Ressource : Illustrationen
%D 2022
%Z Veröffentlicht auf dem Publikationsserver der RWTH Aachen University
%Z Dissertation, Rheinisch-Westfälische Technische Hochschule Aachen, 2022
%X Certain Petrov-Galerkin methods deliver inherently stable formulations of variational problems for a given mesh by selecting appropriate pairs of trial and test spaces. Furthermore, these schemes are especially suited for adaptation due to their inherent ability to yield robust a posteriori error estimates. These numerical methods are also known as Petrov-Galerkin schemes (PG schemes) with optimal test functions. On the other hand, metric-based continuous mesh models have previously been formulated with the aim to build (near) optimal anisotropic meshes with respect to interpolation error models associated with piecewise polynomial approximation spaces. These are compatible with any numerical scheme using such spaces. The primary goal of this dissertation is to formulate the correct continuous-mesh error models for the optimal Petrov-Galerkin methodology, thus pairing the ability to produce near-optimal anisotropic meshes with a numerical scheme which in turn produces optimal stability and approximation properties on these meshes. Employing discontinuous test and trial spaces with optimal test function framework leads to the well-known discontinuous Petrov-Galerkin (DPG) schemes with optimal test functions. This dissertation presents an anisotropic metric-based mesh adaptation strategy utilizing the inbuilt a posteriori error estimator of the DPG schemes. Thus, we end up coupling DPG schemes with metric-based adaptations to generate (near) optimal approximation spaces. Moreover, employing continuous mesh models provides the flexibility of controlling the computational cost in terms of the degrees of freedom. Furthermore, an hp−adaptation technique based on a continuous mesh model has been proposed in this dissertation. Solution variables do not hold one’s prime interest in many engineering applications. In such cases, certain output functionals that depend upon the solution variables are of much more importance. Examples of such output functionals are aerodynamic drag and flux across a boundary. Hence, making goal-oriented adaptation indispensable for generating optimal meshes for such problems. Thus, the proposed adaptation strategies have been extended to perform goal-oriented mesh adaptations. The goal-oriented adaptation strategy utilizes an error estimator from the DPG-star method. Finally, the construction of near-optimal meshes is demonstrated using problems containing boundary layers and singularities for solution-driven refinements and goal-oriented refinements.
%F PUB:(DE-HGF)11
%9 Dissertation / PhD Thesis
%R 10.18154/RWTH-2022-08228
%U https://publications.rwth-aachen.de/record/852630