Adaptive finite element refinement

The aim of this chapter is to illustrate adaptive finite element refinement. This chapter describes various methods of estimating errors and adaptive refinement. These methods constitute a very important tool for practical application of finite element methods. This chapter covers a large range of applications, and discusses only the relatively simple range of linear elasticity and similar self-adjoint problems. Many different norms or measures of error can be used and for some problems, the energy norm is not in fact natural. This chapter discusses the methods that can be used to reduce the errors once a finite element solution has been obtained. As the process depends on previous results at all stages, it is called adaptive. Various procedures exist for the refinement of finite element solutions. Broadly, these fall into two categories: the h-refinement in which the same class of elements continue to be used but are changed in size, in some locations made larger and in others made smaller, to provide maximum economy in reaching the desired solution; and the p-refinement in which we continue to use the same element size and simply increase, generally hierarchically, the order of the polynomial used in their definition.

[1]  J. Tinsley Oden,et al.  PROGRESS ON PRACTICAL METHODS OF ERROR ESTIMATION FOR ENGINEERING CALCULATIONS , 2001 .

[2]  O. C. Zienkiewicz,et al.  The Finite Element Method for Fluid Dynamics , 2005 .

[3]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[4]  I. Babuška,et al.  The h-p version of the finite element method , 1986 .

[5]  B. Lucier.  A moving mesh numerical method for hyperbolic conservation laws , 1986 .

[6]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[7]  O. C. Zienkiewicz,et al.  The Finite Element Method for Solid and Structural Mechanics , 2013 .

[8]  J. Z. Zhu,et al.  Effective and practical h–p‐version adaptive analysis procedures for the finite element method , 1989 .

[9]  I. Babuska,et al.  The $h{\text{ - }}p$ Version of the Finite Element Method for Domains with Curved Boundaries , 1988 .

[10]  Leszek Demkowicz,et al.  Toward a universal h-p adaptive finite element strategy , 1989 .

[11]  J. Oden,et al.  A discontinuous hp finite element method for convection—diffusion problems , 1999 .

[12]  Manil Suri,et al.  ON THE SELECTION OF A LOCKING‐FREE hp ELEMENT FOR ELASTICITY PROBLEMS , 1997 .

[13]  Ivo Babuška,et al.  Approximation properties of the h-p version of the finite element method , 1996 .

[14]  Ivo Babuška,et al.  The p-Version of the Finite Element Method for Parabolic Equations. Part 1 , 1981 .

[15]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[16]  O. Zienkiewicz,et al.  A new approach to the development of automatic quadrilateral mesh generation , 1991 .

[17]  Keith Miller,et al.  Moving Finite Elements. I , 1981 .

[18]  B. Szabó.  Mesh design for the p-version of the finite element method , 1986 .

[19]  Marco Picasso,et al.  Adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives , 2006 .

[20]  E. R. A. Oliveira.  Theoretical foundations of the finite element method , 1968 .

[21]  Gabriel Bugeda,et al.  A STUDY OF MESH OPTIMALITY CRITERIA IN ADAPTIVE FINITE ELEMENT ANALYSIS , 1993 .

[22]  I. Babuska,et al.  Adaptive approaches and reliability estimations in finite element analysis , 1979 .

[23]  I. Babuska,et al.  Rairo Modélisation Mathématique Et Analyse Numérique the H-p Version of the Finite Element Method with Quasiuniform Meshes (*) , 2009 .

[24]  O. C. Zienkiewicz,et al.  Adaptive techniques in the finite element method , 1988 .

[25]  J. Oden,et al.  hp-Version discontinuous Galerkin methods for hyperbolic conservation laws , 1996 .

[26]  O. C. Zienkiewicz,et al.  Adaptive remeshing for compressible flow computations , 1987 .

[27]  Ivo Babuška,et al.  A posteriori error analysis and adaptive processes in the finite element method: Part I—error analysis , 1983 .

[28]  Anthony T. Patera,et al.  Bounds for Linear–Functional Outputs of Coercive Partial Differential Equations : Local Indicators and Adaptive Refinement , 1998 .

[29]  Peter Monk,et al.  On the p- and hp-extension of Nédélec's curl-conforming elements , 1994 .

[30]  Andrew J. Wathen,et al.  Moving finite element methods for evolutionary problems. II. Applications , 1988 .

[31]  Ivo Babuška,et al.  A posteriori error analysis and adaptive processes in the finite element method: Part II—adaptive mesh refinement , 1983 .

[32]  L. Demkowicz,et al.  hp-adaptive finite elements in electromagnetics , 1999 .