Available online at www.sciencedirect.com Mathematics and Computers in Simulation 78 (2008) 645–652 An anisotropic unstructured triangular adaptive mesh algorithm based on error and error gradient information F. Marcuzzi ∗ , M.Morandi Cecchi, M. Venturin Dipartimento di Matematica Pura ed Applicata, Universit´ a degli Studi di Padova, Via Belzoni 7, 35131 Padova, Italy Available online 11 April 2008 Abstract In this paper, an algorithm based on unstructured triangular meshes using standard refinement patterns for anisotropic adaptive meshes is presented. It consists of three main actions: anisotropic refinement, solution-weighted smoothing and patch unrefinement. Moreover, a hierarchical mesh formulation is used. The main idea is to use the error and error gradient on each mesh element to locally control the anisotropy of the mesh. The proposed algorithm is tested on interpolation and boundary-value problems with a discontinuous solution. © 2008 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Adaptive mesh refinement; Unrefinement; Anisotropic meshes; Finite elements; Error estimation and adaptivity 1. Introduction It is very common, in the numerical modelling of problems arising in continuum mechanics, to use triangulations as a spatial support to obtain a discrete, finite dimensional model. Nearly equilateral triangles form isotropic meshes that are excellent for some applications. For other applications, the presence of elements that have an evident elongation produce anisotropic meshes that offer the same accuracy with fewer elements [2,4]. For example, it is well known in the Computational Fluid Dynamics (CFD) literature that many physical problems are characterized by solutions exhibiting directional features and, in this situation, the effectiveness of the finite element method (FEM) can be improved if the mesh is anisotropic and suitably oriented. In an adaptive (iterative) mesh generation process [15,1], the first step requires an initial adequate meshing of the domain, obtained by Delaunay triangulation [4] or advancing front methods [5]. The solution obtained is analysed through a posteriori error estimators and indicators. Their main role is to localize the part of the domain where it is necessary to do mesh refinements or unrefinements and to determine the size of the new elements. In particular, it is possible to reduce effectively and efficiently the error, refining (or unrefining) the area of the mesh where the error estimate is higher (or lower) with respect to a certain threshold. Then, a new mesh is produced and the entire process is repeated. In the approach, here presented, unstructured triangles with local h refinement are used. In this paper, we will consider recovery based error estimators [16,17] because they can give ‘directional’ information about the error (see Section 2), which can be used by the mesh adaptation algorithm (see Section 3) to produce an anisotropic mesh. Comparison results are given between the algorithm here proposed and standard refinement strategies, i.e. regular refinement [3] and longest edge bisection [12]. ∗ Corresponding author. E-mail addresses: marcuzzi@math.unipd.it (F. Marcuzzi), mcecchi@math.unipd.it (M.Morandi Cecchi), venturin@math.unipd.it (M. Venturin). 0378-4754/$32.00 © 2008 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2008.04.006