Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete $H^1$ norm best approximation error estimates for $H^2$ functions hold for arbitrary triangulations. However, similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on the example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily slow speed. The results complement analogous findings for conforming P1 elements.Comment: 23 pages, 10 figure

Schwarz Iterative Methods: Infinite Space Splittings

We prove the convergence of greedy and randomized versions of Schwarz iterative methods for solving linear elliptic variational problems based on infinite space splittings of a Hilbert space. For the greedy case, we show a squared error decay rate of $O((m+1)^{-1})$ for elements of an approximation space $\mathcal{A}_1$ related to the underlying splitting. For the randomized case, we show an expected squared error decay rate of $O((m+1)^{-1})$ on a class $\mathcal{A}_{\infty}^{\pi}\subset \mathcal{A}_1$ depending on the probability distribution.Comment: Revised version, accepted in Constr. Appro

Stochastic subspace correction in Hilbert space

We consider an incremental approximation method for solving variational problems in infinite-dimensional Hilbert spaces, where in each step a randomly and independently selected subproblem from an infinite collection of subproblems is solved. we show that convergence rates for the expectation of the squared error can be guaranteed under weaker conditions than previously established in [Constr. Approx. 44:1 (2016), 121-139]. A connection to the theory of learning algorithms in reproducing kernel Hilbert spaces is revealed.Comment: 15 page

Composite primal/dual √3-subdivision schemes

We present new families of primal and dual subdivision schemes for triangle meshes and 3-refinement. The proposed schemes use two simple local rules which cycle between primal and dual meshes a number of times. The resulting surfaces become very smooth at regular vertices if the number of cycles is ⩾2. The C^1-property is violated only at low-valence irregular vertices, and can be restored by slight modifications of the local rules used. As a generalization, we introduce a wide class of composite subdivision schemes suitable for arbitrary topologies and refinement rules. A composite scheme is defined by a simple upsampling from the coarse to a refined topology, embedded into a cascade of geometric averaging operators acting on coarse and/or refined topologies. We propose a small set of such averaging rules (and some of their parametric extensions) which allow for the switching between control nets associated with the same or different topologic elements (vertices, edges, faces), and show a number of examples, based on triangles, that the resulting class of composite subdivision schemes contains new and old, primal and dual schemes for 3-refinement as well as for quadrisection. As a common observation from the examples considered, we found that irregular vertex treatment is necessary only at vertices of low valence, and can easily be implemented by using generic modifications of some elementary averaging rules

Wages, profits and rent-sharing

The paper suggests a new test for rent-sharing in the U.S. labor market. Using an unbalanced panel from the manufacturing sector, it shows that a rise in a sector's profitability leads after some years to an increase in the long-run level of wages in that sector. The paper controls for workers' characteristics, for industry fixed-effects, and for unionism. Lester's range of wages is estimated, for rent-sharing reasons alone, at approximately 24 per cent of the mean wage