Positive basis convergence

[tsokar]

Dear all,
I was playing with different basis types, and I experienced something weird regarding the BasisType::Positive basis: conducting a h convergence study for a given problem (Laplace equation, Dirichlet BCs), I get optimal convergence rates for BasisType::GaussLobatto, but suboptimal for positive basis (see the attached picture).
I do not think it comes from my code as I am using standard mfem functions, and only changing the basis type between the two runs. The error is computed using ComputeGradError, but the behaviour is similar using the L2 norm of the error.

Do you think I missed something in my code, or was it somehow expected ?

Thanks !
T.

[pazner]

If you have inhomogeneous Dirichlet BCs (and use e.g. ProjectBdrCoefficient to evaluate them), this will only be first-order with the positive (Bernstein) basis. The "projection" is done by evaluation of the function at the Bernstein nodes, but because the Bernstein basis is not interpolatory, this is only accurate to $\mathcal{O}(h)$.

[tsokar]

Ouch, thank you !
I somehow remembered that ProjectCoefficient was not doing a full projection for Bernstein, but I forgot that I used ProjectBdrCoefficient in the treatment of the boundary conditions...

Thanks !