[Paper Review] Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations

Norikazu Saito|arXiv (Cornell University)|Oct 29, 2017

Advanced Numerical Methods in Computational Mathematics24 references3 citations

TL;DR

This paper presents a novel variational analysis of the discontinuous Galerkin (dG) time-stepping method for parabolic equations using abstract evolution equations in Hilbert spaces. By establishing a discrete inf-sup condition and leveraging variational formulations, the authors derive nearly best approximation properties and optimal order error estimates for both the dG method and its finite element space-time approximation, simplifying analysis for problems with variable coefficients and arbitrary polynomial degrees.

ABSTRACT

The discontinuous Galerkin (DG) time-stepping method applied to abstract evolution equation of parabolic type is studied using a variational approach. We establish the inf-sup condition or Babuška--Brezzi condition for the DG bilinear form. Then, a nearly best approximation property and a nearly symmetric error estimate are obtained as corollaries. Moreover, the optimal order error estimates under appropriate regularity assumption on the solution are derived as direct applications of the standard interpolation error estimates. Our method of analysis is new for the DG time-stepping method; it differs from previous works by which the method is formulated as the one-step method. We apply our abstract results to the finite element approximation of a second order parabolic equation with space-time variable coefficient functions in a polyhedral domain, and derive the optimal order error estimates in several norms.

Motivation & Objective

• To develop a new variational approach for analyzing the discontinuous Galerkin (dG) time-stepping method for abstract parabolic evolution equations.
• To establish the discrete inf-sup condition for the dG bilinear form, enabling robust error analysis.
• To derive nearly best approximation and nearly symmetric error estimates as corollaries of the inf-sup condition.
• To apply the abstract results to finite element approximations of second-order parabolic equations with space-time variable coefficients.
• To obtain optimal order error estimates in multiple norms under standard regularity assumptions.

Proposed method

• Formulate the parabolic problem as an abstract evolution equation in a Hilbert triplet $V \subset H \subset V'$ with time-dependent, coercive, and bounded operators.
• Define the dG$(q)$ method as a Galerkin approximation in space-time using piecewise polynomials of degree $q$ in time.
• Introduce specialized DG norms depending on time partitions to analyze the discrete inf-sup condition.
• Prove the discrete inf-sup condition using a duality argument and equivalence of norms in the discrete test and trial spaces.
• Apply the continuous and discrete inf-sup conditions to derive nearly best approximation and symmetric error estimates.
• Use standard interpolation error estimates and the abstract error framework to obtain optimal order error bounds in $L^2(J;H)$, $H^1(J;H^{-1})$, and $L^2(J;V)$ norms.

Experimental results

Research questions

• RQ1 Can the dG$(q)$ method be analyzed via a variational framework that avoids the traditional one-step formulation, especially for problems with variable coefficients?
• RQ2 What is the role of the discrete inf-sup condition in ensuring stability and convergence of the dG$(q)$ method for parabolic equations?
• RQ3 How do the nearly best approximation and symmetric error estimates emerge from the inf-sup condition in the abstract setting?
• RQ4 What are the optimal convergence rates for the dG$(q)$ finite element method when applied to second-order parabolic equations with space-time variable coefficients?
• RQ5 Can the abstract variational framework be extended to derive error estimates in multiple norms, including $L^2(J;H)$ and $H^1(J;H^{-1})$?

Key findings

• The discrete inf-sup condition for the dG bilinear form is established, which is fundamental for proving stability and convergence.
• Nearly best approximation and nearly symmetric error estimates are derived as direct consequences of the inf-sup condition.
• Optimal order error estimates of order $O(h^k + \tau^{q+1})$ are obtained in $L^2(J;H)$ and $H^1(J;H^{-1})$ norms under appropriate regularity assumptions.
• The method achieves optimal convergence in $L^2(J;V)$ norm with order $O(h^k + \tau^{q+1})$ for the finite element approximation.
• The analysis simplifies significantly for variable-coefficient problems and arbitrary $q$, as it avoids the complexity of one-step formulation.
• The framework is validated through application to second-order parabolic equations in polyhedral domains with space-time variable coefficients.