Local bisection refinement for nn-simplicial grids generated by reflection

A simple local bisection refinement algorithm for the adaptive refinement of $n$-simplicial grids is presented. The algorithm requires that the vertices of each simplex be ordered in a special way relative to those in neighboring simplices. It is proven that certain regular simplicial grids on $[0,1]^n$ have this property, and the more general grids to which this method is applicable are discussed. The edges to be bisected are determined by an ordering of the simplex vertices, without local or global computation or communication. Further, the number of congruency classes in a locally refined grid turns out to be bounded above by $n$, independent of the level of refinement. Simplicial grids of higher dimension are frequently used to approximate solution manifolds of parametrized equations, for instance, as in [W. C. Rheinboldt, Numer. Math., 53 (1988), pp. 165–180] and [E. Allgower and K. Georg, Utilitas Math., 16 (1979), pp. 123–129]. They are also used for the determination of fixed points of functions from ${\bf R}^n$ to ${\bf R}^n$, as described in [M. J. Todd, Lecture Notes in Economic and Mathematical Systems, 124, Springer-Verlag, Berlin, 1976]. In two and three dimensions, such grids of triangles, respectively, tetrahedrons, are used for the computation of finite element solutions of partial differential equations, for example, as in [O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems, Academic Press, Orlando, 1984], [R. E. Bank and B. D. Welfert, SIAM J. Numer. Anal., 28 (1991), pp. 591–623], [W. F. Mitchell, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 146–147], and [M. C. Rivara, J. Comput. Appl. Math., 36 (1991), pp. 79–89]. The new method is applicable to any triangular grid and may possibly be applied to many tetrahedral grids using additional closure refinement to avoid incompatibilities

Bisecting with optimal similarity bound on 3D unstructured conformal meshes

We propose a new method to mark for bisection the edges of an arbitrary 3D unstructured conformal mesh. For these meshes, the approach conformingly marks all the tetrahedra with coplanar edge marks. To this end, the method needs three key ingredients. First, we propose a specific edge ordering. Second, marking with this ordering, we guarantee that the mesh becomes conformingly marked. Third, we also ensure that all the marks are coplanar in each tetrahedron. To demonstrate the marking method, we implement an existent marked bisection approach. Using this implementation, we mark and then locally refine 3D unstructured conformal meshes. We conclude that the resulting marked bisection features an optimal bound of similarity classes per tetrahedron.This project has received funding from the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme under grant agreement No 715546. This work has also received funding from the Generalitat de Catalunya under grant number 2017 SGR 1731. The work of X. Roca has been partially supported by the Spanish Ministerio de Economía y Competitividad under the personal grant agreement RYC-2015-01633.Peer ReviewedPostprint (published version

Instance optimal Crouzeix-Raviart adaptive finite element methods for the Poisson and Stokes problems

We extend the ideas of Diening, Kreuzer, and Stevenson [Instance optimality of the adaptive maximum strategy, Found. Comput. Math. (2015)], from conforming approximations of the Poisson problem to nonconforming Crouzeix-Raviart approximations of the Poisson and the Stokes problem in 2D. As a consequence, we obtain instance optimality of an AFEM with a modified maximum marking strategy