Main results: H1-stable polynomial extension on a tetrahedron; H(div)-stable polynomial extension on a tetrahedron; stable broken H1 polynomial extension on a patch of thetrahedra; stable broken H(div) polynomial extension on a patch of thetrahedra; polynomial-degree-robust efficiency of H1-nonconforming methods; polynomial-degree-robust efficiency of H(div)-nonconforming methods. Details: paper (preprint) with Alexandre Ern…
Main results (heat equation): inf-sup equality with constant 1; any order conforming continuous Galerkin discretization in space; any order discontinuous Galerkin discretization in time; guranteed reliability (upper bound); efficiency local with respect to both space and time; robustness with respect to polynomial degrees both in space and time; arbitrary space…
Main results: guaranteed upper and lower bounds for an arbitrary simple eigenvalue, multiple eigenvalue, or a cluster of eigenvalues; guaranteed energy error bounds on the associated eigenvector(s); efficiency (optimal convergence rate) and polynomial-degree robustness; multiplicative constant tending to one under an elliptic regularity assumption on the corresponding source problem; inexact…
Main results: dual norm of any bounded linear functional on the Sobolev space W01,p localizes under an orthogonality condition: equals the lq sum of local contributions; estimates taking into account the violation of the orthogonality condition; distance to the Sobolev space H01 localizes: equals the l2 sum of local…
Main results: total, algebraic, and discretization errors: guaranteed upper and lower bounds; valid for arbitrary iterative algebraic solver; estimating the local distribution of the errors over the computational domain; safe stopping criteria for iterative algebraic solvers: the algebraic error will lie below the discretization one; reliability and efficiency; recovering mass…
Main results: guaranteed error upper bound on each DD iteration; distinguishes the domain decomposition error, the spatial discretization error, and possibly the temporal discretization error; important savings in terms of the number of domain decomposition iterations; optimized Robin and Ventcell transmission conditions; global-in-time formulation that allows for use of local…
Main results: potential and equilibrated flux reconstruction estimates in a unified framework; inhomogeneous Dirichlet and Neumann boundary conditions; spatially varying polynomial degree; mixed rectangular-triangular grids; combination with the H(div)-liftings paper: robustness with respect to the number of hanging nodes; asymptotic exactness observed for smooth solutions; hp-adaptivity strategy numerically leading to…
