Least-Squares Galerkin Methods for Parabolic Problems II: The Fully Discrete Case and Adaptive Algorithms

Abstract

In this second part of our series on least-squares Galerkin methods for parabolic initial-boundary value problems we study the full discretization in time and space. 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. Based on the analysis of the semi-discretization in time carried out in the first part, we construct a posteriori error estimators for the approximation error components associated with time and space. For the space discretization error we use a hierarchical basis estimator with respect to the functional minimization problem. We prove a strengthened Cauchy--Schwarz inequality between coarse and hierarchical surplus space and a bound on the condition number of the auxiliary problem, both uniform in the mesh-size h and the time-step $\tau$ implying efficiency and reliability of the error estimator. This allows for an adaptive strategy for both the time-step size and the spatial triangulation, keeping a proper balance between these two components. Numerical experiments illustrate the performance of the resulting adaptive schemes in time and space.