Discretization of the Region of Interest

[EN]The meccano method was recently introduced to construct simultaneously tetrahedral meshes and volumetric parameterizations of solids. The method requires the information of the solid geometry that is defined by its surface, a meccano, i.e., an outline of the solid defined by connected polyhedral pieces, and a tolerance that fixes the desired approximation of the solid surface. The method builds an adaptive tetrahedral mesh of the solid (physical domain) as a deformation of an appropriate tetrahedral mesh of the meccano (parametric domain). The main stages of the procedure involve an admissible mapping between the meccano and the solid boundaries, the nested Kossaczký’s refinement, and our simultaneous untangling and smoothing algorithm. In this chapter, we focus on the application of the method to build tetrahedral meshes over complex terrain, that is interesting for simulation of environmental processes. A digital elevation map of the terrain, the height of the domain, and the required orography approximation are given as input data. In addition, the geometry of buildings or stacks can be considered. In these applications, we have considered a simple cuboid as meccano.Ministerio de Economía y Competitividad, Gobierno de España; Fondos FEDER; Departamento de Educación, Junta de Castilla y León; CONACYT-SENER, Fondo Sectorial CONACYT SENER HIDROCARBUROS

Adaptive Tree Approximation with Finite Element Functions: A First Look

We provide an introduction to adaptive tree approximation with finite element functions over meshes that are generated by bisection. This approximation technique can be seen as a benchmark for adaptive finite element methods, but may be also used therein for the approximation of data and coarsening. Correspondingly, we focus on approximation problems related to adaptive finite element methods, the design and performance of algorithms, and the resulting convergence rates, together with the involved regularity. For simplicity and clarity, these issues are presented and discussed in detail in the univariate case. The additional technicalities and difficulties of the multivariate case are briefly outlined