Preface

Arif Masud
2004 Computer Methods in Applied Mechanics and Engineering  
Preface This volume is dedicated to the papers that were presented at the mini-symposium on "Stabilized and Multiscale Finite Element Methods", organized at the Fifth World Congress on Computational Mechanics. The conference was held at the Technical University in Vienna, Austria, from July 7 to 12, 2002. The scope of the symposium included all aspects of the stabilized and multiscale finite element methods. The papers presented at the symposium included (i) mathematical theory of the
more » ... and multiscale finite element methods, (ii) new stabilized formulations, (iii) application of stabilized methods to multiphysics problems, (iv) implementation issues in stabilized and multiscale methods, and (v) large scale computations with stabilized methods. The underlying philosophy of the stabilized methods is to strengthen the classical variational formulations so that discrete approximations, which would otherwise be unstable, become stable and convergent. The origins of stabilized methods can be traced back to the early 80Õs when Hughes and colleagues realized the issue of lack of stability of the Galerkin method for advection-dominated diffusion problems. In order to correct this deficiency in the standard Galerkin approach they introduced the Streamline-Upwind-Petrov-Galerkin (SUPG) method. Soon thereafter Hughes proposed a generalization of the SUPG method for the Stokes flow problem that opened the door to the mixed-field problems by circumventing the Babuska-Brezzi (BB) inf-sup conditions. The SUPG method turned out to be the forerunner of a new class of stabilization schemes, namely the Galerkin/Least-Squares (GLS) stabilization methods. In the GLS method a least-squares form of the residuals that is based on the corresponding Euler-Lagrange equations is added to the Galerkin finite element formulation. In the context of the advection-dominated diffusion phenomenon it leads to (i) stabilization of the advection operator without upsetting consistency or compromising accuracy, and (ii) circumvention of the BB (inf-sup) condition that restricts the use of many convenient interpolations. During the same era Johnson and coworkers presented the analysis of stabilized methods. A general theory of the stabilized methods was developed and success was achieved on a variety of problems. GLS stabilization was soon followed by the Unusual Stabilized Methods introduced by Franca and coworkers. Concurrently, another class of stabilized methods that is based on the idea of augmenting the Galerkin method with virtual bubble functions was introduced by Brezzi and coworkers. In the mid 90Õs Hughes revisited the origins of the stabilization schemes from a variational multiscale view point and presented the Variational Multiscale Method. In this method the different stabilization techniques come together as special cases of the underlying sub-grid scale modeling concept. The key idea in the HughesÕ Variational Multiscale method is to perform a mathematical nesting of the fine scales into the coarse scales, thereby providing a robust framework wherein all the important features of the total solution are consistently represented in the computed solution. The papers presented in this volume address multiscale issues in computational fluid dynamics, application of the variational multiscale method to unstructured meshes, space-time finite element techniques, time integration schemes, and stabilized methods in solid mechanics. The paper by Gravemier, Wall and 0045-7825/$ -see front matter Ó
doi:10.1016/j.cma.2004.01.003 fatcat:33jowznkmzgjtl5hmnh4iw4zai