Element-wise a posteriori estimates based on hierarchical bases for non-linear parabolic problems

International Journal for Numerical Methods in Engineering

Volumn 63, Issue 8, 2005, Pages 1146-1173

  de Frutos, Javier ^a,b   Novo, Julia ^c  

^a UNIVERSITY OF VALLADOLID   (Spain)

^b HEC MONTRÉAL   (Canada)

^c UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

Author keywords

A posteriori error estimation; Non linear parabolic equations; P version of the finite element method

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; ESTIMATION; NONLINEAR EQUATIONS; PROBLEM SOLVING;

ELLIPTIC PROBLEM; ERROR ESTIMATION; NON-LINEAR PARABOLIC DIFFERENTIAL EQUATION; POSTERIORI ESTIMATE;

ERROR ANALYSIS;

FINITE ELEMENT METHOD; NON-LINEAR SYSTEM;

EID: 21344439543     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/nme.1310     Document Type: Article

References (27)

1. Verfürth R. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley: Chichester, Teubner, Stuttgart, 1996.
2. Ainsworth M, Oden JT. A Posteriori Error Estimation in Finite Element Analysis. Wiley: New York, 2000.
3. Ainsworth M, Senior B. Aspects of an adaptive hp-finite element method: adaptive strategy, conforming approximation and efficient solvers. Computer Methods in Applied Mechanics and Engineering 1997; 150: 65-87.
4. Oden JT, Denkowicz L, Rachowicz W, Westermann TA. Toward a universal h-p adaptive finite-element strategy. Part 2. A posteriori error estimation. Computer Methods in Applied Mechanics and Engineering 1989; 77:113-180.
5. Bernardi C. Indicateurs d'erreur en h-N version des elements spectraux. M2AN 1996; 30: 1-38.
6. Melenk JM, Wohlmuth BI. On residual-based a posteriori error estimation in h-p FEM. A posteriori error estimation and adaptive computational methods. Advances in Computational Mathematics 2001; 15:311-331.
7. Moore P. A posteriori error estimation with finite element semi and fully discrete methods for non-linear parabolic equations in one space dimension. SIAM Journal on Numerical Analysis 1994; 31: 149-169.
8. Adjerid S, Belguendouz B, Flaherty JE. A posteriori finite element error estimation for diffusion problems. SIAM Journal on Scientific Computing 1999; 21:728-746.
9. Babuška I, Ohnimus S. A posteriori error estimation for the semidiscrete finite element method of parabolic differential equations. Computer Methods in Applied Mechanics and Engineering 2001; 190: 4691-4712.
10. Babuška I, Feistauer M, Šolín P. On one approach to a posteriori error estimates for evolution problems solved by the method of lines. Numerische Mathematik 2001; 89:225-256.
11. Stroubulis T, Babuska I, Datta DK. Guaranteed a posteriori error estimation for fully discrete solutions of parabolic problems. International Journal for Numerical Methods in Engineering 2003; 56:1243-1259.
12. Makridakis C, Nochetto RH. Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM Journal on Numerical Analysis 2003; 41:1585-1594.
13. de Frutos J, Novo J. A posteriori error estimation with the p version of the finite element method for non-linear parabolic differential equations. Computer Methods in Applied Mechanics and Engineering 2002; 191:4893-4904.
14. Bank RE, Smith RK. A posteriori error-estimates based on hierarchical bases. SIAM Journal on Numerical Analysis 1993; 30: 921-935.
15. de Frutos J, Novo J. A postprocess based improvement of the spectral element method. Applied Numerical Mathematics 2000; 33: 217-223.
16. de Frutos J, Novo J. A spectral element method for the Navier-Stokes equations with improved accuracy. SIAM Journal on Numerical Analysis 2000; 38:799-819.
17. Garcí-Archilla B, Titi ES. Postprocessing the Galerkin method: the finite element case. SIAM Journal on Numerical Analysis 2000; 37:470-499.
18. de Frutos J, Novo J. Postprocessing the linear finite element method. SIAM Journal on Numerical Analysis 2002; 40:805-819.
19. Henry D. Geometric Theory of Semilinear Parabolic Equations. Springer: Berlin, 1991.
20. Temam R. Infinite dimensional dynamical systems in mechanics and physics. Applied Mathematical Sciences, vol. 68. Springer: New York, 1988.
21. Babuška I, Griebel M, Pitkaranta J. The problem of selecting the shape functions for a p-type finite element. International Journal for Numerical Methods in Engineering 1989; 28:1891-1908.
22. Babuška I, Miller A. A feedback finite element method with a posteriori error estimations. Part I. The finite element method and some basis properties of the a posteriori error estimator. Computer Methods in Applied Mechanics and Engineering 1987; 61: 1-40.
23. Morin P, Nochetto RH, Siebert KG. Local problems on stars: a posteriori error estimators, convergence and performance. Mathematics of Computation 2003; 72:1067-1097.
24. Prudhomme S, Nobile F, Chamoin L, Oden JT. Analysis of a subdomain-based error estimator for finite element approximations of Elliptic problems. Numerical Methods for Partial Differential Equations 2004; 20: 165-192.
25. Eijkhout V, Vassilevski P. The role of the strengthened Cauchy-Buniakowskii-Schwarz inequality in multilevel methods. SIAM Review 1991; 33:405-419.
26. Schwab Ch. p- and hp-finite element methods. Theory and Applications in Solid and Fluid Mechanics. Clarendon Press: Oxford, 1998.
27. Calvo MP, de Frutos J, Novo J. Linearly implicit Runge-Kutta methods for advection-reaction-diffusion equations. Applied Numerical Mathematics 2001; 37:535-549.