A Fully Discrete Adaptive Scheme for Parabolic Equations

Abstract

We present an adaptive algorithm for solving linear parabolic equations using hierarchical B-splines and the implicit Euler method for the spatial and time discretizations, respectively. Our development improves upon one from 2018 from Gaspoz and collaborators, where fully discrete adaptive schemes have been analyzed within the framework of classical finite elements. Our approach is based on an a posteriori error estimation that essentially consists of four indicators: a time and a consistency error indicator that dictate the time-step size adaptation, and coarsening and a space error indicator that are used to obtain suitably adapted hierarchical meshes (at different time-steps). Even though we use hierarchical B-splines for the space discretization, a straightforward generalization to other methods, such as FEM, is possible. The algorithm is guaranteed to reach the final time within a finite number of operations, and keep the space-time error below a prescribed tolerance. Some numerical tests document the practical performance of the proposed adaptive algorithm.