Doctor's Theses (authored and supervised):

C. Rojik:
"p-version projection based interpolation";
Supervisor, Reviewer: J. Melenk, L. Demkowicz, E. Stefan; Institut für Analysis und Scientific Computing, 2020; oral examination: 2020-01-21.

English abstract:
In this thesis, we define p-version projection-based interpolation operators on the reference tetrahedron K. These are defined on the spaces H^2(K), H^1(K,curl)$ and H^(1/2)(K,div), and we show that they are projections onto polynomial spaces, that they satisfy a commuting diagram property and that they have suitable approximation properties, when increasing the polynomial degree. Additionally, the trace of the interpolant on the boundary is fully determined by the trace of the function, which allows the construction of interpolation operators on a grid in an element-wise fashion by transformation of the operators on the reference element. Projection-based interpolation operators were introduced by L. Demkowicz and several coworkers. These operators have optimal approximation properties (as p to infinity) up to logarithmic factors. In this thesis, the logarithmic factor is removed. The regularity requirement is, however, stronger than in the work of Demkowicz. We also get interpolation error estimates in negative Sobolev norms. In 2D, the weakest possible negative norm is here determined by the maximal regularity for solutions of the Poisson problem, since we use duality arguments. In 3D, we also obtain estimates in negative norms. Here we use the fact that the convexity of tetrahedra allows more regularity for the Poisson problem. In this work, we also analyze regularity of solutions for the Poisson equation -\Delta u = f on polygons Omega in 2D, both for Dirichlet and Neumann boundary conditions ("Shift theorem"). Denoting omega the interior angle of Omega at a corner, we show the shift theorem for functions in Sobolev spaces (for a right-hand side f that is in H^(-1+s) with 0 \leq s < pi/omega near a corner, the solution satisfies u\in H^(1+s) locally), however, the focus lies on the limit case s=pi/omega which holds in terms of Besov spaces: Assume Pi/omega \notin N, then f\in B_{2,1}^{-1+\pi/\omega} admits regularity B_{2,\infty}^{1+\pi/\omega}. This result is similar to those shown by Bacuta and Bramble where multilevel theory was used to prove the limit case. However, these results are formulated for functions in non-standard Besov spaces, whereas we use the Mellin calculus which leads to a proof in standard Besov spaces.

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