hp-Optimal DG Approximation and Robust Schwarz Decompositions on One-Irregular Cubical Meshes

We study hp approximation and additive Schwarz decompositions for variable-order cubical finite element spaces on one-irregular meshes.

For fitted homogeneous diffusion interface problems on one-irregular hexahedral meshes, we prove an hp-optimal energy-norm estimate for the interior penalty DG method.

The interpolation input is a conforming hp interpolant obtained from fitted conforming closures of one-irregular vertex patches.

We also derive stable decompositions for conforming and DG spaces.

On one-irregular quadrilateral meshes the bounds allow locally comparable variable polynomial degrees and are independent of the mesh size, the local degrees, and, under a local coefficient quasi-monotonicity condition, the coefficient contrast.

On one-irregular hexahedral meshes the conforming decomposition has the corresponding polylogarithmic loss; the DG-to-conforming reduction is used there for uniform-degree DG spaces.

Numerical experiments illustrate the p-optimal DG error estimate and the robustness of the DG Schwarz preconditioner.