Mesh and Solver Co-adaptation in Finite Element Methods for Anisotropic Problems Qiang Du, 1,2 Zhaohui Huang, 3 Desheng Wang 4,5 1 Department of Mathematics, Penn State University, University Park, PA 16802 2 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China 3 Center for Space Science Research, Chinese Academy of Sciences, Beijing 100080, China 4 Department of Mathematics, Xiangtan University, Xiantan, Hunan 411105, China 5 Civil and Computational Engineering Centre, School of Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK Received 16 July 2004; accepted 15 January 2005 Published online 4 March 2005 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.20072 Mesh generation and algebraic solver are two important aspects of the finite element methodology. In this article, we are concerned with the joint adaptation of the anisotropic triangular mesh and the iterative algebraic solver. Using generic numerical examples pertaining to the accurate and efficient finite element solution of some anisotropic problems, we hereby demonstrate that the processes of geometric mesh adaptation and the algebraic solver construction should be adapted simultaneously. We also propose some techniques applicable to the co-adaptation of both anisotropic meshes and linear solvers. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 21: 859 – 874, 2005 Keywords: finite element; anisotropic meshing; iterative solver; joint co-adaptation I. INTRODUCTION The finite element solution of partial differential equations involves the mesh generation and optimization, the assembly of discrete algebraic systems using the finite element basis, and the solution of these systems by some algebraic solvers. It is well known that the performance of finite element computations depends critically on both the geometric meshes and the algebraic Correspondence to: Qiang Du, Department of Mathematics, Penn State University, University Park, PA 16802 (e-mail: qdu@math.psu.edu) Contract grant sponsor: Chinese State Major Basic Research Project; contract grant number: G1999032800 Contract grant sponsor: U.S. National Science Foundation; contract grant number: DMS-0196522. © 2005 Wiley Periodicals, Inc.