" 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 enoug."
I'll will search in the literature to get deeper information, so thanks for the hint! And the nonlinear equations of elasticity are actually elliptic equations? One last question regarding the convergence rate: As I previously said in this chat, I am projecting the dim*dim coefficients of the stress tensor, which are only availabe at the qps, to the nodes using a global L2-projection and for comparison a superconvergent approach (superconvergent patch recovery introduced by Zienkiewicz and Zhu). I will then also determine the convergence rate of the two approaches. As the name suggests, the superconvergent approach should give me a higher convergence rate *as expected. * -Is the *expected rate* the rate which the laplace equation (assuming that this holds for elasticity as just discussed) yields for the gradients of the solution, i.e. ||grad (u-u_h)||_L2 <= ...h^p ? For me it's not yet fully clear how convergence rate of a projection method (L2, SPR) is linked to ||grad (u-u_h)||_L2 <= ...h^p . Basically the input for the projection methods are the gradients (stresses) at the qps. Of course these values will be changed by the method( apply projection -> get_function_values), but is this "change" totally independent of the rate p from ||grad (u-u_h)||_L2 <= ...h^p ? For me this seems to be the case since in the literature people compare the rate of a L2-projection with the rate of a SPR-projection (or other methods). Thanks again for helping me! Best Simon Am Mo., 7. Juni 2021 um 17:47 Uhr schrieb Wolfgang Bangerth < bange...@colostate.edu>: > 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. > > -- > ------------------------------------------------------------------------ > Wolfgang Bangerth email: bange...@colostate.edu > www: http://www.math.colostate.edu/~bangerth/ > > -- > The deal.II project is located at http://www.dealii.org/ > For mailing list/forum options, see > https://groups.google.com/d/forum/dealii?hl=en > --- > You received this message because you are subscribed to the Google Groups > "deal.II User Group" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to dealii+unsubscr...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/e751f64b-5cc3-d823-0855-5e51d6dd4d46%40colostate.edu > . > -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to dealii+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/CAM50jEuZUUR0U65WcYWXYynGkt8GfnBDOvHEZh96jNR8FXPTNg%40mail.gmail.com.
