Anisotropic finite elements for the Stokes problem: a posteriori error estimator and adaptive mesh

AbstractWe propose an a posteriori error estimator for the Stokes problem using the Crouzeix–Raviart/P0 pair. Its efficiency and reliability on highly stretched meshes are investigated. The analysis is based on hierarchical space splitting whose main ingredients are the strengthened Cauchy–Schwarz inequality and the saturation assumption. We give a theoretical proof of a method to enrich the Crouzeix–Raviart element so that the strengthened Cauchy constant is always bounded away from unity independently of the aspect ratio. An anisotropic self-adaptive mesh refinement approach for which the saturation assumption is valid will be described. Our theory is confirmed by corroborative numerical tests which include an internal layer, a boundary layer, a re-entrant corner and a crack simulation. A comparison of the exact error and the a posteriori one with respect to the aspect ratio will be demonstrated

ON THE GENERATION OF HIERARCHICAL MESHES FOR MULTILEVEL FEM AND BEM SOLVERS FROM CAD DATA

As numerical techniques for solving PDE or integral equations become more sophisticated, treatments of the generation of the geometric inputs should also follow that numerical advancement. This document describes the preparation of CAD data so that they can later be applied to hierarchical BEM or FEM solvers. For the BEM case, the geometric data are described by surfaces which we want to decompose into several curved foursided patches. We show the treatment of untrimmed and trimmed surfaces. In particular, we provide prevention of smooth corners which are bad for diffeomorphism. Additionally, we consider the problem of characterizing whether a Coons map is a diffeomorphism from the unit square onto a planar domain delineated by four given curves. We aim primarily at having not only theoretically correct conditions but also practically efficient methods. As for FEM geometric preparation, we need to decompose a 3D solid into a set of curved tetrahedra. First, we describe some method of decomposition without adding too many Steiner points (additional points not belonging to the initial boundary nodes of the boundary surface). Then, we provide a methodology for efficiently checking whether a tetrahedral transfinite interpolation is regular. That is done by a combination of degree reduction technique and subdivision. Along with the method description, we report also on some interesting practical results from real CAD data

The ITL programming interface toolkit

This document serves as a reference for the beta version of our evaluation library ITL. First, it describes a library which gives an easy way for programmers to evaluate the 3D image and the normal vector corresponding to a parameter value which belongs to the unit square. The API functions which are described in this document let programmers make those evaluations without the need to understand the underlying CAD complica- tions. As a consequence, programmers can concentrate on their own scien- tific interests. Our second objective is to describe the input which is a set of parametric four-sided surfaces that have the structure required by some integral equation solvers

Geometric processing of CAD data and meshes as input of integral equation solvers

Among the presently known numerical solvers of integral equations, two main categories of approaches can be traced: mesh-free approaches, mesh-based approaches. We will propose some techniques to process geometric data so that they can be efficiently used in subsequent numerical treatments of integral equations. In order to prepare geometric information so that the above two approaches can be automatically applied, we need the following items: (1) Splitting a given surface into several four-sided patches, (2) Generating a diffeomorphism from the unit square to a foursided patch, (3) Generating a mesh M on a given surface, (4) Patching of a given triangulation. In order to have a splitting, we need to approximate the surfaces first by polygonal regions. We use afterwards quadrangulation techniques by removing quadrilaterals repeatedly. We will generate the diffeomorphisms by means of transfinite interpolations of Coons and Gordon types. The generation of a mesh M from a piecewise Riemannian surface will use some generalized Delaunay techniques in which the mesh size will be determined with the help of the Laplace-Beltrami operator. We will describe our experiences with the IGES format because of two reasons. First, most of our implementations have been done with it. Next, some of the proposed methodologies assume that the curve and surface representations are similar to those of IGES. Patching a mesh consists in approximating or interpolating it by a set of practical surfaces such as B-spline patches. That approach proves useful when we want to utilize a mesh-free integral equation solver but the input geometry is represented as a mesh

The ITL programming interface toolkit

This document serves as a reference for the beta version of our evaluation library ITL. First, it describes a library which gives an easy way for programmers to evaluate the 3D image and the normal vector corresponding to a parameter value which belongs to the unit square. The API functions which are described in this document let programmers make those evaluations without the need to understand the underlying CAD complica- tions. As a consequence, programmers can concentrate on their own scien- tific interests. Our second objective is to describe the input which is a set of parametric four-sided surfaces that have the structure required by some integral equation solvers