Least-Squares Galerkin Methods for Parabolic Problems I: Semidiscretization in Time

Abstract

This is the first part of a series of papers on least-squares Galerkin methods for parabolic initial-boundary value problems. These methods are based on the minimization of a least-squares functional for an equivalent first-order system over space and time with respect to suitable discrete spaces. This paper presents the derivation and analysis of one-step methods for semi-discretization in time from least-squares principles for linear parabolic problems. One of the most important features of the least-squares methodology is the built-in a posteriori estimate for the approximation error. This is a consequence of the equivalence of the least-squares functional to the consistency error associated with a time-step, measured in an appropriate norm. For the presentation in this paper, we focus our attention on the specific combination of piecewise linear, not necessarily continuous, functions with continuous piecewise linear functions for the flux and scalar variable, respectively. For the resulting method, a convergence result is shown for the scalar variable and the associated flux.