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

Abstract

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.

Withdrawn by the authors. The current version requires substantial revision to clarify the assumptions and proof of the main decomposition theorem. The authors will prepare a revised version