Pointwise best approximation results for Galerkin finite element solutions of parabolic problems

In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty$ norm. The discretization method uses of continuous Lagrange finite elements in space and discontinuous Galerkin methods in time of an arbitrary order. The method of proof differs from the established fully discrete error estimate techniques and for the first time allows to obtain such results in three space dimensions. It uses elliptic results, discrete resolvent estimates in weighted norms, and the discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in [16]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish a local best approximation property that shows a more local behavior of the error at a given point

Maximum–norm a posteriori error estimates for an optimal control problem

We analyze a reliable and efficient max-norm a posteriori error estimator for a control-constrained, linear–quadratic optimal control problem. The estimator yields optimal experimental rates of convergence within an adaptive loop. © 2019, Springer Science+Business Media, LLC, part of Springer Nature

Maximum norm error estimates for an isoparametric finite element discretization of elliptic boundary value problems

In this paper, we consider an elliptic boundary value problem on a domain with regular boundary and discretize it with isoparametric finite elements of order $k\geq1$. We show optimal order of convergence of the isoparametric finite element solution in the $W^{1,\infty}$-norm. As an intermediate step, we derive stability and convergence estimates of optimal order $k$ for a (generalized) Ritz map

Optimal W1,∞^{1,∞}-estimates for an isoparametric finite element discretization of elliptic boundary value problems

In this paper, we consider an elliptic boundary value problem on a domain with regular boundary and discretize it with isoparametric finite elements of order $k\geq1$. We show optimal order of convergence of the isoparametric finite element solution in the $W^{1,\infty}$-norm. As an intermediate step, we derive stability and convergence estimates of optimal order $k$ for a (generalized) Ritz map