Global and interior pointwise best approximation results for the gradient of Galerkin solutions for parabolic problems
Provides rigorous error bounds for finite element/dG discretizations of parabolic PDEs, benefiting numerical analysts working on error estimation for time-dependent problems.
The paper proves global and interior pointwise best approximation results for the gradient of fully discrete Galerkin solutions to parabolic problems, achieving optimal error estimates in the L^∞(I;W^{1,∞}(Ω)) norm without requiring mesh-size/time-step coupling.
In this paper we establish best approximation property of fully discrete Galerkin solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Om))$ norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in \cite{LeykekhmanD_VexlerB_2016b}. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish interior best approximation property that shows more local dependence of the error at a point.