Dear Ameachi,

Hello and many thanks for this nice question. It seems to me that the response depends on the trend of responses convergence and the rate of convergence. Provided proper convergence, Richardson extrapolation proposed long ago leads to a response very close to the exact response. With two responses w1 and w2 obtained with number of elements equal to n1 and n2 in a 2D problem, considering 1/n1^0.5 and 1/n2^0.5 as the elements sizes respectively, and assuming first order of accuracy, implementation of the Richardson extrapolation based on two approximately computed responses seemingly results in the first alternative in the attached file, and considering second order of accuracy leads to (n2w2-n1w1)/(n2-n1) which is different from both of the alternatives in the PDF file attached; see also the reference below:

Richardson LF, Gaunt JA. The deferred approach to the limit. Philosophical Transactions of the Royal Society of London 1927; 226:299–361.

The paper below

Soroushian, A., Wriggers, P., and Farjoodi, J. (2009). “Asymptotic upperbounds

for the errors of Richardson Extrapolation with practical application

in approximate computations.” Int. J. Numer. Methods Eng., 80(5), 565–595.

and its references on proper convergence may also help.

With best wishes looking forward to the continuation of the discussion...

Hello Aram Soroushian

 Thanks for your answer. Be rest assured that the 2nd order extrapolation you gave namely w∞ =  (n2w2-n1w1)/(n2-n1) is exactly the same as my own Eq. (2b) namely  

         w∞ =  w2 +  (w2-w1)/(n2/n1 - 1)          (2b)

as you can verify by multiplying the numerator and denominator of the fraction by n1 and adding to w2.

Basically, I am surprised at the results obtained by Fraser and Wardle (1976) in FEA of a plate on a halspace, that both displacement and bending moment converge linearly in accordance with Eq. (1). Text books on FEM have insinuated that in a displacement based FEA, the displacements converge quadratically (2nd order) while the Bending moments, obtained from 2nd derivatives of displacements , converge linearly (1st order) in accordance with Eq. (2b). Thus, Fraser and Wardle seem to have contradicted supposedly established ideas, and their proposition was based on results of numerous analyses.   

Fraser, R.A. and Wardle, L.J. (1976) Numerical analysis of rectangular rafts on layered foundation, Geotechnique, 26(4), 613-630.

Hello Aram Soroushian

In the first instance Can I get from you a copy of your paper

Soroushian, A., Wriggers, P., and Farjoodi, J. (2009). “Asymptotic upperbounds

for the errors of Richardson Extrapolation with practical application

in approximate computations.” Int. J. Numer. Methods Eng., 80(5), 565–595.

In the 2nd instance you mention that " that the response depends on the trend of responses convergence and the rate of convergence."  Very good given three results 17.337, 18.803, 19.292 obtained using 16, 64 and 256 elements respectively: What is the order of the convergence, that is if the solution varies as 1/nα what is α? 

An alternative approach to improve FEA is to view FEA as the leading approximation to an in principle exact closure of the discretisation model.  With some more algebra one  constructs better approximations without refining the mesh.   See the following paper for the ideas and references to earlier developments: A. J. Roberts, T. MacKenzie, and J. Bunder. A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions. J. Engineering Mathematics, 86(1):175–207, 2014.

In 1D problems my web service does the construction on any in a wide domain of problems you might enter, see  http://www.maths.adelaide.edu.au/anthony.roberts/holistic1.php

Dear Ameachi and J.L.Roberts,

Sorry for not writing earlier; I had and have to finish something soon. 

Richardson extrapolation is a fast way towards the exact solution and its mere competitors are seemingly the Richardson extrapolation of higher orders, i.e. those computed using more computed solutions. Nevertheless, for successful implementation of Richardson extrapolation the computed solutions need to converge properly. This implies that the changes of errors with respect to the algorithmic parameter in a log-log plot need to be as a line with a slope close to a positive integer number (equal to the theoretical rate of convergence). Of course based on the formal definition of error, the above test is not practical. Instead you can use pseudo-errors and depict the log-log plot of pseudo-errors with respect to the algorithmic parameter (i.e. pseudo convergence plot) and check whether the changes are as a straight line with an integer slope. (It is almost proved that the convergence and pseudo convergence plots both display proper convergence or both display improper convergence). With your numbers, the algorithmic parameter, which is here the length of the elements in the 2D mesh, is 0.25, 0.125, and 0.0625, and from the three values of the solution we can come to only two values of the pseudo errors 1.466 and 0.489. Hence, in a pseudo convergence plot we will surely have a straight line, but the slope is considerably more than one (and we are not sure about the theoretical rate of convergence). For implementing Richardson extrapolation however you need an integer value for the slope, which can generally be computed separately from theory or numerically in a simple example. However, the non-integer slope implies that your computed solutions are not in their proper convergence region; when at the ideally proper convergence region (the slope is precisely a positive integer) the result of Richardson extrapolation is precisely equal to the exact solution; otherwise, some errors exist that are obtainable from the second reference introduced in my previous answer to this question. In any way an assumption in all this discussion from the most beginning is the uniform changes of the sizes of elements throughout the 2D domain, i.e. when you change the number of elements from N to 4N, the size of all elements halve. On the proper convergence and pseudo error and pseudo convergence let me introduce you to my papers below:

1. Soroushian, A. (2010b). “Proper convergence, a concept new in science and important in engineering.” Proc., 4th Int. Conf. From Scientific Computing to Computational Engineering, Laboratory of Fluid Mechanics and Energy, Athens, Greece.

2. Soroushian, A. (2010c). “Pseudo convergence and its implementation in engineering approximate computations.” Proc., 4th Int. Conf. From Scientific Computing to Computational Engineering, Laboratory of Fluid Mechanics and Energy, Athens, Greece.

and the recent extension of the second paper above you can see in my research gate contributions.

And many thanks from Dr. Roberts for introducing a nice paper. By the way a book on extensions is as noted below:

Brezenski C, Redivo Zagalia M. Extrapolation Methods: Theory and Practice. North-Holland: Amsterdam, 1991.

Again many thanks for the nice question causing this discussion. Have a nice day, nicer weekend, and great future.

With Kind Regards,

Aram