A FULLY DISCRETE ADAPTIVE SCHEME FOR PARABOLIC EQUATIONS

EDUARDO M. GARAU; FERNANDO D. GASPOZ; PEDRO MORIN; RAFAEL VÁZQUEZ

Congreso; XXIV Congreso sobre Métodos Numéricos y sus Aplicaciones; 2019

We present an adaptive algorithm for solving linear parabolic equations using hierarchicalB-splines and the implicit Euler method for the spatial and time discretizations, respectively. Our developmentimproves upon one from 2018 from Gaspoz and collaborators, where fully discrete adaptiveschemes have been analyzed within the framework of classical finite elements. Our approach is basedon an a posteriori error estimation that essentially consists of four indicators: a time and a consistencyerror indicator that dictate the time-step size adaptation, and coarsening and a space error indicator thatare used to obtain suitably adapted hierarchical meshes (at different time-steps). Even though we usehierarchical 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 ofoperations, and keep the space-time error below a prescribed tolerance. Some numerical tests documentthe practical performance of the proposed adaptive algorithm.