On 6/7/21 8:12 AM, Simon wrote:
Yes, actually both schemes converge to the same solution.
Maybe I should do some postprocessing with a visualiuation in order to figure
out what the lumped integration does with the values at the qps.
Then the difference you observe is simply numerical error.
I want to determine the convergence rate for the elasticity equations with the
PDE: div(stress-tensor) = 0.
-Since I determine the convergence rate, the before mentioned *constant* in
the inequality cancels out, i.e. I should get the same convergence rate as for
the laplace equation. Is that right?
Correct. The *rate* is independent of the factor.
-I actually solve the nonlinear elasticity equations (hyperelasticity); The
finite element spaces, test functions,... are the same as for linear
elasticity, but the stresses depend now nonlinear on the gradient of the
displacements. My question is if the nonlinearity changes the error estimates?
Let´s assume I determine the convergence rate in the L2-norm for linear
elasticity, which is in the best case 2 for for p=1. Should I also get 2 for
the nonlinear elasticity equations?
I'm sure someone has proved this, but my intuition is that for every elliptic
equation without an advection term, you will get the same convergence rate as
for the Laplace equation as long as the coefficient in the elliptic operator
is nice enough. Nice enough here will mean that your stress-strain
relationship does not degenerate, i.e.,
||sigma|| / ||eps||
does not go to zero or infinity, and is a smooth function of some sort.
Best
W.
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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