Optimal auxiliary-variable convergence for general Hodge–Laplace form degrees

Determine whether the order- (r+\frac{1}{2}) convergence rate for the auxiliary variables in the degree- k Hodge–Laplace HDG method can be sharpened to the optimal order- (r+1) rate for general k, beyond the cases k=1 and k=n-1.

Background

For the degree-k Hodge–Laplace problem, the paper proves order-(r+1) convergence for the k-form solution, while the auxiliary variables generally achieve only order-(r+\frac{1}{2}) convergence. The analysis improves this auxiliary-variable rate to order-(r+1) when k=1 and/or k=n-1, but does not establish the optimal rate for general form degrees.

The authors note that optimal-order estimates are already known for the scalar Poisson cases k=0 and k=n, using HDG projection and M-decomposition techniques. The unresolved problem is whether analogous techniques can be extended to the general Hodge–Laplace setting, potentially also yielding superconvergence and postprocessing results.