Abstract:
Given a scalar-valued discontinuous piecewise polynomial on a simplicial mesh, a potential reconstructionis a piecewise polynomial that is trace continuous, i.e., H1-conforming. It is bestobtained via local homogeneous Dirichlet problems (finite element primal energy minimization) on vertexpatches. Similarly, given a vector-valued discontinuous piecewise polynomial not having the target curlor divergence, a flux reconstruction is a piecewise polynomial that is tangential- or normal-tracecontinuous, i.e., H(curl)- or H(div)-conforming, and has the target curl or divergence. It is bestobtained via local homogeneous Neumann problems (finite element dual energy constrained minimization).The theory of broken polynomial extension operators [1, 2] allows to prove that such discrete,piecewise polynomial minimizations are equivalent to the continuous (Sobolev spaces) ones uniformlyin the polynomial degree p. In a second step, a local p-stable decomposition such as [Falk & Winther,Local space-preserving decompositions for the bubble transform, Found. Comput. Math. 26 (2025),1219–1267] may be employed and some developments from algebraic topology (simplicial complexes,shellability, subdivisions) may be used.The concepts of potential and flux reconstructions are known to lead to guaranteed, locally efficient,and polynomial-degree-robust a posteriori error estimates, which we quickly recall here.We focus on showing that potential and flux reconstructions can also be used to obtain novel apriori error estimates. They in particular allow to devise local commuting projectors [3, 4, 5]that are uniformly stable in the polynomial degree p, with the approximation property being thatof local-best unconstrained projection. This means that the precision of the commuting projectoris, up to a constant only possibly depending on the mesh shape regularity, as good as that of theL2 projection over individual mesh elements, without any continuity requirement along mesh faces, andwithout any curl or divergence constraint. Therefrom, optimal hp approximation estimates, where theminimal Sobolev regularity is only requested separately on each mesh element, follow. The designed localcommuting projectors naturally allow treatment of nontrivial domains and partial boundary conditions.

16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2

ETH Zürich

Seminar on Stochastic Processes

Prof. Dr. Rajat Subhra Hazra (University of Leiden)

Opinion dynamics on random networks: Consensus and rewiring abstract

17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43

Thursday, November 5

ETH Zürich

Nachdiplomvorlesung

Kaibo Hu (University of Oxford)

Finite Element Tensor Calculus

10:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43