Martin Vohralik, Professor, Inria Paris and CERMICS, Ecole nationale des ponts et chaussées, France

Wei Gong, Professor & Shipeng Mao, Professor & Weiying Zheng, Professor

Potential and flux reconstructions for optimal a priori and a posteriori error estimates

10:00-11:00 May 30(Friday), N702

Given a scalar-valued discontinuous piecewise polynomial, a “potential reconstruction” is a scalar-valued piecewise polynomial that is (trace-)continuous, i.e., H1-conforming. It is best obtained via a solution of local homogeneous Dirichlet problems on patches of elements sharing a vertex, using the conforming finite element method. Similarly, given a vector-valued discontinuous piecewise polynomial not satisfying the target divergence, an “equilibrated flux reconstruction” is a vector-valued piecewise polynomial that is normal-trace-continuous, i.e., H(div)-conforming, which possesses the target divergence. It is best obtained via a solution of local homogeneous Neumann problems on patches of elements, using the mixed finite element method. These concepts are known to provide guaranteed, locally efficient, and polynomial-degree-robust a posteriori error estimates for numerical approximations of model partial differential equations. We show that they also allow to devise stable local commuting projectors. These lead to polynomial-degree-robust equivalence of global continuous and local discontinuous approximation and optimal hp approximation / a priori error estimates under minimal elementwise Sobolev regularity.

Demkowicz, L., and Vohralík, M. p-robust equivalence of global continuous constrained and local discontinuous unconstrained approximation, a p-stable local commuting projector, and optimal elementwise hp approximation estimates in H(div). HAL Preprint 04503603, submitted for

publication, 2024.

Ern, A., Gudi, T., Smears, I., and Vohralík, M. Equivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div). IMA J. Numer. Anal. 42, 2 (2022), 1023–1049.

Ern, A., and Vohralík, M. A posteriori error estimation based on potential and flux reconstruction for the heat equation. SIAM J. Numer. Anal. 48, 1 (2010), 198–223.

Ern, A., and Vohralík, M. Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53, 2 (2015), 1058–1081.

Ern, A., and Vohralík, M. Stable broken H1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions. Math. Comp. 89, 322 (2020), 551–594.

Vohralík, M. p-robust equivalence of global continuous and local discontinuous approximation, a p-stable local projector, and optimal elementwise hp approximation estimates in H1. HAL Preprint 04436063, submitted for publication, 2024.