Least-Squares Galerkin Methods for Parabolic Problems III: Semidiscrete Case for Semilinear Problems

Abstract

This is the third 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 semidiscretization in time from least-squares principles for semilinear parabolic problems. One of the most important features of the least-squares methodology is a built-in a posteriori estimate for the approximation error. For the presentation in this paper, we focus our attention on the specific combination of piecewise linear, not necessarily continuous, functions in time with continuous piecewise linear for the flux and scalar variables, respectively. For the resulting method, a convergence result is shown for the scalar variable.