Space-time adaptivity for parabolic evolution equations

Jan Westerdiep (University of Amsterdam)

Taking the well-posed mixed simultaneous space-time variational formulation introduced in [And13], we use methods previously developed in [SW20] to construct an adaptive loop that produces space-time approximations as linear combinations of tensor-products of wavelets in time and finite elements in space.

Using an efficient and reliable ‘hierarchical basis’ error estimator, we apply bulk chasing to show convergence of the iterands. Moreover, we provide an algorithm for linear-complexity application of the system matrix and an optimal preconditioner. Lastly, we include an extensive numerical study to show that this method is competitive in terms of speed and moreover exhibits optimal convergence rate in the number of degrees of freedom.


[And13] R. Andreev. Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal., 33(1):242–260, 2013.

[SW20] Stability of Galerkin discretizations of a mixed space-time variational formulation of parabolic evolution equations. IMA J. Numer. Anal., 2020.