Adaptivity for the heat equation using hierarchical splines

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

Pavia

Congreso; V International Conference on Isogemetric Analysis; 2017

International Center for Numerical Methods in Engineering (CIMNE)

We present an adaptive algorithm for solving the heat equation using hierarchical splines and theimplicit Euler method for the spatial and time discretizations, respectively. Our developmentfollows closely the lines in (Kreuzer et al, 2012; Gaspoz et al 2017), where fully discrete adaptive schemes have been analyzed withinthe framework of classical finite elements.Our approach is based on an a posteriori error estimation that essentially consists of four indi-cators. On the one hand, we have a time error indicator and a consistency error indicator thatdictate the time-step size adaptation. On the other hand, we have a coarsening error indicatorand a space error indicator that are used to obtain a suitably adapted hierarchical mesh (at eachfixed discrete time). In particular, the considered space error indicators are the function-baseda posteriori error estimators for elliptic problems introduced in (Buffa & Garau, 2016). The algorithm is guaranteedto reach the final time within a finite number of operations, and keeps the the space-time errorbelow a prescribed tolerance.We finally present some numerical tests to illustrate the performance of the proposed adaptivealgorithm, using the implementation of hierarchical splines presented in (Garau & Vazquez, 2017).