Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

Abstract

Abstract. 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 ∞ (I; W 1,∞ (Ω)) 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 [15]. 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.

Fully discrete best approximation type estimates in $L^{\infty}(I;L^2(Ω)^d)$ for finite element discretizations of the transient Stokes equations

“…In the case of the heat equation, a similar estimate with respect to L ∞ (I; L 2 (Ω )) is derived in Meidner et al (2011) and for a non-autonomous parabolic problem in (Leykekhman & Vexler, 2018, Theorem 4.5). For corresponding estimates in the maximum norm in the case of the heat equation we refer to Eriksson & Johnson (1995); Leykekhman & Vexler (2016); Schatz et al (1980); Meidner et al (2011) and for the maximum norm of the gradient to Leykekhman & Vexler (2017b); Leykekhman & Wahlbin (2008); Thomée et al (1989). Further results are also available in case of discretization only in space.…”

“…Further results are also available in case of discretization only in space. For an overview and respective references we refer to Leykekhman & Vexler (2016, 2017b.…”

Discrete Maximal Parabolic Regularity for Galerkin Finite Element Methods for Nonautonomous Parabolic Problems

“…Such results are very useful, for example, in a priori error estimates and essential in obtaining error estimates where the spatial mesh size h and the time steps k are independent of each other (cf. [29,30]).…”

Global and local pointwise error estimates for finite element approximations to the Stokes problem on convex polyhedra

“…This paper is the first paper in our program to establish best approximation results for the fully discrete approximations for transient Stokes systems in L ∞ and W 1,∞ norms. Similar program was carried out by the last two authors for the parabolic problems in a series of papers [15,16,17,18]. The approach there relies on stability of the Ritz projection, resolvent estimates in L ∞ and W 1,∞ norms and discrete maximum parabolic regularity.…”