Exponential Convergence of hp-Time-Stepping in Space-Time Discretizations of Parabolic PDEs

@article{Perugia2022ExponentialCO,
  title={Exponential Convergence of hp-Time-Stepping in Space-Time Discretizations of Parabolic PDEs},
  author={Ilaria Perugia and Christoph Schwab and Marco Zank},
  journal={ArXiv},
  year={2022},
  volume={abs/2203.11879}
}
For linear parabolic initial-boundary value problems with selfadjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains D, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove… 

Figures and Tables from this paper

Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation
We present quadrature schemes to calculate matrices, where the so-called modified Hilbert transformation is involved. These matrices occur as temporal parts of Galerkin finite element discretizations

References

SHOWING 1-10 OF 44 REFERENCES
Petrov–Galerkin space-time hp-approximation of parabolic equations in H1/2
  • D. Devaud
  • Mathematics
    IMA Journal of Numerical Analysis
  • 2019
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
Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces
TLDR
A space-time variational formulation of parabolic initial-boundary value problems in anisotropic Sobolev spaces in combination with a Hilbert-type transformation and new efficient direct solvers for this system are proposed and investigated.
hp-Version Space-Time Discontinuous Galerkin Methods for Parabolic Problems on Prismatic Meshes
TLDR
An extensive comparison among the new space-time dG method, the (standard) tensorized space- time dG methods, the classical dG-time-stepping, and conforming finite element method in space, is presented in a series of numerical experiments.
Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method
TLDR
The discontinuous Galerkin finite element method (DGFEM) for the time discretization of parabolic problems is analyzed in the context of the hp-version of theGalerkin method and it is shown that the hp's spectral convergence gives spectral convergence in problems with smooth time dependence.
Space-time least-squares finite elements for parabolic equations
Coercive space-time finite element methods for initial boundary value problems
Abstract. We propose and analyse new space-time Galerkin-Bubnov-type finite element formulations of parabolic and hyperbolic second-order partial differential equations in finite time intervals.
Stability of Galerkin discretizations of a mixed space–time variational formulation of parabolic evolution equations
TLDR
Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs are analyzed and the resulting Galerkin operators are shown to be uniformly stable.
Space-Time Approximation with Sparse Grids
TLDR
This article introduces approximation spaces, especially suited for the approximation of solutions of parabolic problems, which are based on the tensor product construction of a multiscale basis in space and a multiracial basis in time and gives numerical results on the convergence rate of the interpolation error of these spaces for various space-time problems with two spatial dimensions.
Further results on a space-time FOSLS formulation of parabolic PDEs
TLDR
Well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation is proven, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated.
ANALYSIS OF THE FINITE ELEMENT METHOD FOR TRANSMISSION/MIXED BOUNDARY VALUE PROBLEMS ON GENERAL POLYGONAL DOMAINS ∗
We study theoretical and practical issues arising in the impl ementation of the Finite Element Method for a strongly elliptic second order equation with jump discontinuities in its coefficients on a
...
...