A posteriori error analysis for finite element solution of one-dimensional elliptic differential equations using equidistributing meshes

The paper is concerned with the adaptive linear finite element solution of linear one-dimensional elliptic differential equations using equidistributing meshes. A strategy is developed for defining such meshes based on a residual-based a posteriori error estimate. The mesh and the finite element solution are determined by the coupled system of the finite element equation and the equidistribution relation (i.e., the mesh equation). An iterative algorithm is proposed for solving this system for the mesh and the finite element solution. The existence of an equidistributing mesh is proven for a given sufficiently large number of the points with help of a result on the continuous dependence of the finite element solution on the mesh, which is also established in the current work. Error bounds for the finite element solution are obtained for the equidistributing and quasi-equidistributing meshes. They show that adaptive meshes can lead to more accurate solutions than a uniform mesh and it is unnecessary to compute the equidistribution relation accurately for the equidistributing meshes. The departure from the equidistributing meshes has only a mild effect on the finite element error. Numerical examples are given to illustrate the convergence of the iterative algorithm and the theoretical findings.

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

[2]  Ivo Babuška,et al.  Analysis of optimal finite-element meshes in ¹ , 1979 .

[3]  F. Moukalled,et al.  A local adaptive grid procedure for incompressible flows with multigridding and equidistribution concepts , 1991 .

[4]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[5]  J. A. Mackenzie.  Uniform Convergence Analysis of an Upwind Finite-Difference Approximation of a Convection-Diffusion , 1996 .

[6]  Long Chen,et al.  Convergence and optimality of adaptive mixed finite element methods , 2010, Math. Comput..

[7]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[8]  Rob P. Stevenson,et al.  An Optimal Adaptive Finite Element Method , 2004, SIAM J. Numer. Anal..

[9]  Ricardo H. Nochetto,et al.  Data Oscillation and Convergence of Adaptive FEM , 2000, SIAM J. Numer. Anal..

[10]  Tao Tang,et al.  Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: analysis of convergence , 2000 .

[11]  I. Babuska,et al.  Analysis of Optimal Finite Element Meshes in R1 , 1979 .

[12]  George Beckett,et al.  Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem , 2000 .

[13]  H. G. Burchard,et al.  Splines (with optimal knots) are better , 1974 .

[14]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

[15]  Long Chen,et al.  Stability and accuracy of adapted finite element methods for singularly perturbed problems , 2008, Numerische Mathematik.

[16]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[17]  Victor Pereyra,et al.  Mesh selection for discrete solution of boundary problems in ordinary differential equations , 1974 .

[18]  Weizhang Huang,et al.  Variational mesh adaptation II: error estimates and monitor functions , 2003 .

[19]  R. Bank,et al.  Some a posteriori error estimators for elliptic partial differential equations , 1985 .

[20]  J. A. White,et al.  On Selection of Equidistributing Meshes for Two-Point Boundary-Value Problems , 1979 .

[21]  J. Mackenzie,et al.  On a uniformly accurate finite difference approximation of a singulary peturbed reaction-diffusion problem using grid equidistribution , 2001 .

[22]  Robert D. Russell,et al.  Convergence of de Boor's algorithm for the generation of equidistributing meshes , 2011 .

[23]  J. Sacks,et al.  Designs for Regression Problems with Correlated Errors III , 1966 .

[24]  Carl de Boor,et al.  Good approximation by splines with variable knots + , 2011 .

[25]  D. M. Sloan,et al.  Analysis of difference approximations to a singularly perturbed two-point boundary value problem on an adaptively generated grid , 1999 .

[26]  C. D. Boor,et al.  Good approximation by splines with variable knots. II , 1974 .

[27]  Christian Kreuzer,et al.  Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method , 2008, SIAM J. Numer. Anal..

[28]  Natalia Kopteva,et al.  A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem , 2001, SIAM J. Numer. Anal..

[29]  J. Sacks,et al.  Designs for Regression Problems With Correlated Errors: Many Parameters , 1968 .

[30]  John D. Pryce.  On the convergence of iterated remeshing , 1989 .

[31]  Long Chen,et al.  Optimal anisotropic meshes for minimizing interpolation errors in Lp-norm , 2007, Math. Comput..

[32]  Weizhang Huang,et al.  Metric tensors for anisotropic mesh generation , 2005 .

[33]  P. George,et al.  Delaunay mesh generation governed by metric specifications. Part I algorithms , 1997 .

[34]  George Beckett,et al.  Uniformly convergent high order finite element solutions of a singularly perturbed reaction-diffusion equation using mesh equidistribution , 2001 .

[35]  Weizhang Huang,et al.  Variational mesh adaptation: isotropy and equidistribution , 2001 .

[36]  Carsten Carstensen,et al.  Error reduction and convergence for an adaptive mixed finite element method , 2006, Math. Comput..

[37]  Marc Beckers,et al.  The construction of three-dimensional invariant cubature formulae , 1991 .

[38]  Weizhang Huang.  Practical aspects of formulation and solution of moving mesh partial differential equations , 2001 .

[39]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[40]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[41]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods , 2002, SIAM Rev..

[42]  Rüdiger Verführt,et al.  A review of a posteriori error estimation and adaptive mesh-refinement techniques , 1996, Advances in numerical mathematics.

[43]  D. S. Dodson.  Optimal-order approximation by polynomial spline functions. , 1972 .

[44]  S. SIAMJ..  MEASURING MESH QUALITIES AND APPLICATION TO VARIATIONAL MESH ADAPTATION , 2005 .