Petrov-Galerkin space-time hp-approximation of parabolic equations in H1/2

We analyse a class of variational space-time discretizations for a broad class of initial boundary value problems for linear, parabolic evolution equations. The space-time variational formulation is based on fractional Sobolev spaces of order 1/2 and the Riemann-Liouville derivative of order 1/2 with respect to the temporal variable. It accommodates general, conforming space discretizations and naturally accommodates discretization of infinite horizon evolution problems. We prove an inf-sup condition for hp-time semidiscretizations with an explicit expression of stable test functions given in terms of Hilbert transforms of the corresponding trial functions; inf-sup constants are independent of temporal order and the time-step sequences, allowing quasi-optimal, high-order discretizations on graded time-step sequences, and also hp-time discretizations. For solutions exhibiting Gevrey regularity in time and taking values in certain weighted Bochner spaces, we establish novel exponential convergence estimates in terms of Nt, the number of (elliptic) spatial problems to be solved. The space-time variational setting allows general space discretizations and, in particular, for spatial hp-FEM discretizations. We report numerical tests of the method for model problems in one space dimension with typical singular solutions in the spatial and temporal variable. hp-discretizations in both spatial and temporal variables are used without any loss of stability, resulting in overall exponential convergence of the space-time discretization. © The Author(s) 2019. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.