Minimal residual space-time discretizations of parabolic equations: Asymmetric spatial operators

TL;DR

This paper develops a minimal residual space-time discretization method for parabolic equations that remains stable and accurate even when the spatial operator is asymmetric, broadening applicability.

Contribution

It introduces a stable discretization approach for parabolic equations that does not require symmetry of the spatial operator, with proven quasi-optimality and error estimates.

Findings

Quasi-optimality of the discretization under LBB stability.

Error estimates in energy-norm independent of spatial operator symmetry.

Applicability to asymmetric spatial operators in parabolic equations.

Abstract

We consider a minimal residual discretization of a simultaneous space-time variational formulation of parabolic evolution equations. Under the usual `LBB' stability condition on pairs of trial- and test spaces we show quasi-optimality of the numerical approximations without assuming symmetry of the spatial part of the differential operator. Under a stronger LBB condition we show error estimates in an energy-norm which are independent of this spatial differential operator.