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Inf-sup stable space-time Local Discontinuous Galerkin method for the heat equation.

Sergio Gómez1,2, Chiara Perinati3, Paul Stocker4

  • 1Department of Mathematics and Applications, University of Milano-Bicocca, 20125 Milan, Italy.

Journal of Scientific Computing
|December 8, 2025
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Summary
This summary is machine-generated.

We introduce a new space-time Local Discontinuous Galerkin method for parabolic problems. This method proves solution existence and uniqueness, offering robust error bounds and validated convergence rates for various polynomial spaces.

Keywords:
Inf-sup stabilityLocal Discontinuous Galerkin methodParabolic problemPrismatic space–time meshesSpace–time finite element method

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • Parabolic problems are fundamental in modeling various physical phenomena.
  • Existing numerical methods often face limitations with complex geometries or specific solution properties.
  • Efficient and accurate approximation techniques are crucial for solving these equations.

Purpose of the Study:

  • To develop and analyze a novel space-time Local Discontinuous Galerkin (LDG) method.
  • To establish theoretical foundations for the method's stability and accuracy.
  • To demonstrate its applicability to a wide range of discrete spaces and meshes.

Main Methods:

  • Formulation of a space-time LDG method for parabolic equations.
  • Proof of existence and uniqueness of discrete solutions using inf-sup conditions.
  • Derivation of hp-a priori error bounds.
  • Analysis of convergence rates for different polynomial spaces.

Main Results:

  • The proposed space-time LDG method is rigorously analyzed.
  • Existence and uniqueness of discrete solutions are proven without relying on polynomial inverse estimates.
  • A second inf-sup condition provides control over the time derivative.
  • hp-a priori error bounds are derived, confirming convergence rates.

Conclusions:

  • The developed space-time LDG method offers a robust framework for approximating parabolic problems.
  • The theoretical results are validated by numerical experiments.
  • The method demonstrates flexibility with general discrete spaces and prismatic meshes.