Combined Error Estimates in the Case of Dimension Reduction

Abstract. We consider the stationary reaction-diffusion problem in a domain Ω⊂ℝ 3 $\Omega \subset \mathbb {R}^3$ having the size along one coordinate direction essentially smaller than along the others. By an energy type argumentation, different simplified models of lower dimension can be deduced and solved numerically. For these models, we derive a guaranteed upper bound of the difference between the exact solution of the original problem and a three-dimensional reconstruction generated by the solution of a dimensionally reduced problem. This estimate of the total error is determined as the sum of discretization and modeling errors, which are both explicit and computable. The corresponding discretization errors are estimated by a posteriori estimates of the functional type. Modeling error majorants are also explicitly evaluated. Hence, a numerical strategy based on the balancing modeling and discretization errors can be derived in order to provide an economical way of getting an approximate solution with an a priori given accuracy. Numerical tests are presented and discussed.

[1]  Sergey Repin,et al.  Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces , 2012 .

[2]  Philippe G. Ciarlet,et al.  Plates And Junctions In Elastic Multi-Structures , 1990 .

[3]  S. Sauter,et al.  A Posteriori Estimation of Dimension Reduction Errors , 2004 .

[4]  Sergey Repin,et al.  Functional Approach to Locally Based A Posteriori Error Estimates for Elliptic and Parabolic Problems , 2006 .

[5]  Douglas N. Arnold,et al.  Derivation and Justification of Plate Models by Variational Methods , 2000 .

[6]  Ivo Babuška,et al.  On a dimensional reduction method. II. Some approximation-theoretic results , 1981 .

[7]  Ivo Babuška,et al.  A posteriori error estimation for hierarchic models of elliptic boundary value problems on thin domains , 1996 .

[8]  S. Sauter,et al.  Estimates of the modeling error for the Kirchhoff–Love plate model , 2010 .

[9]  I. Babuska,et al.  On a dimensional reduction method. I. The optimal selection of basis functions , 1981 .

[10]  Sergey Repin,et al.  Functional a posteriori estimates for the reaction–diffusion problem , 2006 .

[11]  S. Repin,et al.  Estimates of dimension reduction errors for stationary reaction–diffusion problems , 2011 .

[12]  Philippe G. Ciarlet,et al.  A Justi cation of a Nolinear Model in Plate Theory , 1979 .

[13]  Pekka Neittaanmäki,et al.  Reliable Methods for Computer Simulation: Error Control and a Posteriori Estimates , 2004 .

[14]  S. Repin.  A Posteriori Estimates for Partial Differential Equations , 2008 .