Speaker
Description
In this work, we revisit the results of Babuška and Suri [1] in the setting of mesh elements that are star-shaped with respect to a ball, with the goal of deriving hp-approximation estimates featuring fully computable multiplicative constants. To this end, we establish in particular an H1 extension theorem with completely explicit stability bound. This further allows us to derive Poincaré–Friedrichs-type inequalities for L2 polynomial projections with explicit dependence on the geometric parameters and on the polynomial degree.
Building on these refined estimates, we devise a reliable residual-type a posteriori error estimator for the Hybrid High-Order (HHO) discretization of the Poisson problem which is fully explicit with respect to h, p, and the chunkiness parameter ϱ. The resulting bound is robust with respect to the polynomial degree and provides a quantitative foundation for hp-adaptive strategies on general (star-shaped) polytopal meshes.
Our results help bridge the gap between the abstract theory on general meshes (see, e.g., [2]) and practical adaptive implementations that require fully computable bounds. They may also be of independent interest for other nonconforming methods based on L2 polynomial projections, such as weak Galerkin schemes.
References
[1] I. Babuška and M. Suri. The optimal convergence rate of the p-version of the finite element method. SIAM Journal on Numerical Analysis, 1987.
[2] D. A. Di Pietro and J. Droniou. The Hybrid High-Order Method for Polytopal Meshes. Springer, Cham, 2020. Modeling, Simulation and Applications, Vol. 19