Are FE computations accurate; really?
 

I have been 'practising' the FE method as a 'code developer' for 13 years, and have a reasonable knowledge of the underlying theory (convergence, error analysis etc). Yet i would very much like to know what 'experts' in FE would have to sau about the followimg problem:

(P) solve a linear (incompressible) stokes flow problem

driven by gravity . Relax a gaussian (or sinusoidal) free surface for example .

Trivial problem would you say. Maybe. Let me ask the following questions first. Try to answer them before you look at the section below which is a one sentence summary of my observations.

(mu=1 h=1 L=infinite or periodic K=infinite. rau*g=-1) amplitude=a .

1. what is the analytical solution when a-->0 2. before running any experiment would a smart FE expert know exactly what grid to design to achieve a 1% error in infinite (=max) norm

well this list doesnt have to be much longer.

My observations:

I have tried 4 types of quad elements and 3 different (really different!) implementations of Crouzeix Raviart elements on this problem. I get 2 answers. 2 consistent answers (all Qs: pian sumihara1 Qe6 Q41 pian sumihara2 give one answer, all T6=CR another. Uzwawa improvements on CR confirm the first iteration)

This is quite puzzling. Do not blame it too rapidly on a bug. Fundamental solutions for some problems (radial half space with dirac source at origin) have been passed (another interesting result) at 1.d-14 max norm error (with significant sampling near the singularity) for Q41 and .. 1.d-9 for CR (!)...

Ok you may say i say nothing about the grid. Fine. I have tools to make grids , and do delaunay adaptive computations. But really the problem posed is a 'large-wavelength' problem. I used very modest grids of order 200x50. Uniform refinement is not changing the picture significantly.

You may also try a slope stability problem for which again the Q and T elements behave quite differently. In fact even the topology of velocity contours is different.

I conjecture that this type of 'Neumann' problem is hard. I find it unfortunate that FE litterature is content with trivial tests and unpractical theory..I would be very happy to hear that i am wrong.

Can anyone convince me one way or the other?

Philippe Fullsack Research assistant Dalhousie University Halifax Nova-Scotia CANADA

P.S. This question originates from the many tests i am performing with our FE program Mozart, developed in the Dalhousie Geophysics team under Professor Chris Beaumont. We are interested in the integration in time of creeping flows with strong nonlinearities and thermal dependence. Further 'complications' involve strain softening / hardening, and development of anisotropies. I take the view that linear predictors should be well understood if we wish to call our simulations 'quantitative' .