#### Conference abstracts

Session B6 - Multiresolution and Adaptivity in Numerical PDEs

July 14, 17:00 ~ 17:25

## A posteriori error estimates in $L^2(H^1)\cap H^1(H^{-1})$ and $L^2(H^1)$-norms for high-order discretizations of parabolic problems

### Iain Smears

### Inria Paris, France - iain.smears@inria.fr

We present a posteriori error estimates in parabolic energy norms for space-time discretisations based on arbitrary-order conforming FEM in space and arbitrary-order discontinuous Galerkin methods in time. Using the heat equation as a model problem, we first consider a posteriori error estimates in a norm of $L^2(H^1)\cap H^1(H^{-1})$-type that is extended to the nonconforming discrete space. We construct a flux equilibration by solving at each time-step some local mixed finite element problems posed on the patches of the mesh, which yields estimators with guaranteed upper bounds on the error, and locally space-time efficient lower bounds with respect to this extended norm. Furthermore, the efficiency constants are robust with respect to the discretisation parameters, including the polynomial degrees in both space and time, and also with respect to refinement and coarsening between time-steps, thereby removing the need for the transition conditions required in earlier works. In the last part of the talk, we consider the same flux equilibration in the context of $L^2(H^1)$-norm a posteriori error estimation, where we show that in the practically relevant situation where the time-step size $\tau \gtrsim h^2$ the mesh-size, then the spatial part of the estimators have the additional feature of being locally efficient with respect to the $L^2(H^1)$-norm of the error plus the temporal jumps. Our analysis of efficiency in $L^2(H^1)$-norms thus removes the more severe time-step and mesh size conditions required previously in the literature.

Joint work with Alexandre Ern (ENPC, France) and Martin Vohralik (Inria Paris, France).