diff --git a/dual-problems.tex b/dual-problems.tex index 6e0f82b..6dcd8ae 100644 --- a/dual-problems.tex +++ b/dual-problems.tex @@ -373,8 +373,8 @@ \subsection{Verification on solvable test cases} The verification exercises we use are taken from those used to test the implementation of the primal form of SSA in the icepack package\citep{shapero2021icepack}. We compare numerical results on a sequence of grids to exactly solvable instances of SSA and check that the results converge with the expected order of accuracy. -Finite element theory predicts that the $L^2$-norm difference between the exact solution and the solutions obtained using $CG(k)$ finite elements is $\mathscr{O}(\delta x^{p + 1})$ where $\delta x$ is the mesh spacing. -If the slope in a log-log fit of error against mesh spacing deviates significantly from $p + 1$, this would indicate some mis-specification of the problem or bug in the solver. +Finite element theory predicts that the $L^2$-norm difference between the exact solution and the solutions obtained using $CG(k)$ finite elements is $\mathscr{O}(\delta x^{k + 1})$ where $\delta x$ is the mesh spacing. +If the slope in a log-log fit of error against mesh spacing deviates significantly from $k + 1$, this would indicate some mis-specification of the problem or bug in the solver. The first test is to use the exact solution for the velocity of a floating ice shelf with thickness \begin{equation}