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

Abstract

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.

References in corpus (1)

• Discrete maximal parabolic regularity for Galerkin finite element methods

Cited by in corpus (9)

• Discrete maximal parabolic regularity for Galerkin finite element methods
• Optimal Error Estimates for Fully Discrete Galerkin Approximations of Semilinear Parabolic Equations
• Discrete maximal parabolic regularity for Galerkin finite element methods for non-autonomous parabolic problem s
• Global and interior pointwise best approximation results for the gradient of Galerkin solutions for parabolic problems
• Maximum norm error estimates for the finite element approximation of parabolic problems on smooth domains
• Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations
• Weak discrete maximum principle of finite element methods in convex polyhedra
• Fully discrete best approximation type estimates in $L^{\infty }(I;L^2(\Omega {)}^d)$ for finite element discretizations of the transient Stokes equations
• Fully Discrete Pointwise Smoothing Error Estimates for Measure Valued Initial Data