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Space-time hp-approximation of parabolic equations in H 1/2

by D. Devaud

(Report number 2018-01)

Abstract
We analyze 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 accomodates general, conforming space discretizations and naturally accomodates discretization of infinite horizon evolution problems. We prove an inf-sup condition for \(hp\)-time semidiscretizations with an explicit expression of stable testfunctions 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 quasioptimal, 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 \(N_t\), 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 variable, are used without any loss of stability, resulting in overall exponential convergence of the space-time discretization.

Keywords: Parabolic partial differential equations, space-time approximation, hp-refinements, exponential convergence, a priori error estimates.

@Techreport{D18_755,
  author = {D. Devaud},
  title = {Space-time hp-approximation of parabolic equations in H 1/2},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-01.pdf },
  year = {2018}
}

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