$h_2$ denote a coarse and a refined mesh size, respectively. From Table \[tbl:convRect\] and Figure \[fig:convRect\], we observe that the proposed HDG method has an optimal convergence order which is $p+1$ for HDG-$\mathbb{P}_p$.

    \[tbl:convRect\]                                                                                        
  -------------------- ---------------------- ------- ---------------------- ------- ---------------------- -------
                                                                                                            
          $h$                  error           order          error           order          error           order
    $5\time10^{-8}$     $1.67\times10^{-9}$      -     $4.52\times10^{-10}$     -     $2.04\times10^{-11}$     -
   $2.5\time10^{-8}$    $4.10\times10^{-10}$    2.0    $5.61\times10^{-11}$    3.0    $1.28\times10^{-12}$    4.0
   $1.25\time10^{-8}$   $9.98\times10^{-11}$    2.0    $7.52\times10^{-12}$    3.0    $7.78\times10^{-14}$    4.0
   $6.25\time10^{-9}$   $2.40\times10^{-12}$    2.1    $9.11\times10^{-13}$    3.0    $5.03\times10^{-15}$    4.0

  : Convergence results for the cavity problem.

![Convergence history of the proposed HDG method for the cavity problem.[]{data-label="fig:convRect"}](./convRect.pdf)

Benchmark problem: a cylindrical plasmonic nanowire
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As our benchmark problem we consider the plasmonic behavior of a cylindrical nanowire. This has been used as a convenient benchmark problem for other numerical methods before [@ToscanoEtal2012; @HiremathEtal2012] because analytical solutions exist both for the local and for the NHD models, see the derivation in Ref. [@Ruppin2001]. We make use of the fact that the analytical Mie solution of Ref. [@Ruppin2001] allows making the nonlocal parameter $\beta$ complex-valued. This enables us to benchmark our HDG simulations against exact analytical results for the GNOR model as well. (For comparison, optical properties of a sphere in the GNOR model, based 