On 3/23/19 7:23 AM, Konrad wrote: > > I am trying to solve a simple elliptic problem (imagine some sort of a > simple Laplace problem for now) with pure Neumann BCs, say -\Delta u = > f, n\grad u = g. Now theory tells me that the volume average over f must > be the same as the boundary average over g. If I solve that problem > (with deal.ii) in the standard form above the Neumann condition is > natural and enters the right hand side. I solve and I see what I expect. > Fine, intuition confirmed. > > Now I am solving it in mixed form with RT0-DGQ0 elements which are > stable. My Neumann BC becomes essential and enters as hard constraint > into the system but the compatibility condition should be the same. When > I solve that system now I see things that totally do not match what I > see in the standard form and I do not have a clue why.
Precisely what do you see? > I can see from tests that also the standard form is very sensitive to > little mismatches in the compatibility condition so I was wondering if > this issue becomes more of a problem in mixed form. I haven't heard of this, but that doesn't mean anything. How do you actually derive the compatibility condition in the mixed case? Let me call the primary variable 'p'. Assume you have Neumann boundary conditions dp/dn = g on the entire boundary Gamma. In the non-mixed, standard case, after multiplying the equation and integrating by parts, you have (nabla q, nabla p)_Omega = (q,f)_Omega + (q,g)_Gamma for all test functions q. You get the compatibility condition by testing with q=1. But in the mixed formulation, your weak formulation is (v,u)_Omega - (div v, p)_Omega - (q, div u)_Omega = -(q,f)_Omega As you correctly mention, 'g' doesn't appear in the weak formulation. I'm pretty sure the compatibility condition is still true (of course!) but I'm not sure how to derive it in the mixed case off the top of my head. Maybe going through the steps would help point out where the issue lies. Best W. -- ------------------------------------------------------------------------ Wolfgang Bangerth email: [email protected] 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 [email protected]. For more options, visit https://groups.google.com/d/optout.
