CM: Polynomial-degree-robust a posteriori error estimates in a unified setting and applications
Speaker: Martin Vohralik (INRIA Paris)
Date: June 19, 2017 15:30
Location: SP2 416-2
We review how to bound the error between unknown weak solutions of model partial differential equations and their numerical approximations via fully computable a posteriori estimates. We present a unified framework based on primal-conforming potential reconstructions, obtained via Dirichlet finite element problems on patches of elements, and on dual-conforming equilibrated flux reconstructions, obtained via patchwise Neumann mixed finite element problems. The framework covers at once all standard numerical methods like the conforming, nonconforming, discontinuous Galerkin, and mixed finite elements. The key for our approach are operators extending piecewise polynomial data prescribed on faces (in the H1 setting) and in addition in the volume (in the H(div) setting). We prove their uniform stability on patches of simplices sharing a vertex in three space dimensions. We detail this key result justifying polynomial-degree-robust efficiency of potential and flux reconstructions in the setting of the model Laplace equation. We finally present recent extensions of this methodology to problems with sign-changing coefficients and to the Laplace eigenvalue, Stokes, and heat problems. Generalization of the methodology to take into account the algebraic error from inexact linear solvers as well as the linearization error from inexact linearizations is described as well. Numerical experiments illustrate the theoretical developments.