[Paper Review] Discrete maximal regularity and the finite element method for parabolic equations

Tomoya Kemmochi, Norikazu Saito|arXiv (Cornell University)|Feb 22, 2016

Advanced Numerical Methods in Computational Mathematics23 references3 citations

TL;DR

This paper establishes a discrete version of maximal regularity for the finite element method applied to linear and semilinear parabolic equations. Using operator-theoretic techniques involving pure imaginary powers of operators, it derives optimal error estimates in $L^p(0,T;L^q( heta))$ norms for mass-lumped finite element schemes, with applications to both linear heat equations and semilinear problems with locally Lipschitz nonlinearities.

ABSTRACT

Maximal regularity is a fundamental concept in the theory of partial differential equations. In this paper, we establish a fully discrete version of maximal regularity for a parabolic equation. We derive various stability results in $L^p(0,T;L^q(Ω))$ norm, $p,q\in (1,\infty)$ for the finite element approximation with the mass-lumping to the linear heat equation. Our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of G.~Dore and A.~Venni (On the closedness of the sum of two closed operators. \emph{Math.\ Z.}, 196(2):189--201, 1987). As an application, optimal order error estimates in that norm are proved. Furthermore, we study the finite element approximation for semilinear heat equations with locally Lipschitz continuous nonlinearity and offer a new method for deriving optimal order error estimates. Some interesting auxiliary results including discrete Gagliardo-Nirenberg and Sobolev inequalities are also presented.

Motivation & Objective

• To develop a discrete version of maximal regularity theory for parabolic equations using the finite element method.
• To derive stability and optimal error estimates in $L^p(0,T;L^q( heta))$ norms for linear and semilinear heat equations.
• To provide a new error analysis framework for semilinear problems with locally Lipschitz nonlinearities, avoiding strong regularity assumptions.
• To establish discrete Gagliardo-Nirenberg and Sobolev inequalities as auxiliary tools for the analysis.

Proposed method

• Employing an operator-theoretic approach based on pure imaginary powers of operators, inspired by Dore and Venni (1987).
• Using the mass-lumping finite element method for spatial semidiscretization of the linear heat equation.
• Defining a fully discrete scheme via Crank-Nicolson or backward Euler time discretization with a weighted average in time (θ-method).
• Applying spectral theory and $H^\infty$-functional calculus to analyze the discrete operator properties.
• Deriving discrete maximal regularity estimates in $L^p(0,T;L^q( heta))$ for $p,q\in(1,\infty)$.
• Establishing optimal error bounds via a new method that avoids requiring global Lipschitz continuity of the nonlinearity in semilinear problems.

Experimental results

Research questions

• RQ1 Can a fully discrete version of maximal regularity be established for the finite element method applied to parabolic equations?
• RQ2 What are the optimal error estimates in $L^p(0,T;L^q( heta))$ norms for linear and semilinear parabolic problems with mass-lumping?
• RQ3 How does the choice of time discretization (e.g., $\theta=0, 1/2, 1$) affect the convergence order in the discrete maximal regularity framework?
• RQ4 Can optimal error estimates be derived for semilinear parabolic equations with only locally Lipschitz continuous nonlinearities using this approach?
• RQ5 What is the role of discrete Gagliardo-Nirenberg and Sobolev inequalities in the error analysis of the finite element scheme?

Key findings

• Optimal error estimates of order $O(h^2 + \tau)$ are established in $L^p(0,T;L^q(\Omega))$ for the mass-lumped finite element method with $\theta = 1/2$ or $\theta = 1$, under appropriate time-step conditions.
• For $\theta = 0$, the method achieves $O(h^2)$ convergence when $\tau = O(h^2)$, consistent with theoretical bounds.
• The scheme (25) yields optimal convergence rates matching theoretical predictions, with observed orders of $O(h^2)$ for $\theta = 1/2$ and $\theta = 1$.
• In contrast, the alternative scheme (97) yields only $O(h)$ error bounds theoretically, but numerical results show $O(h^2)$ convergence in practice, suggesting the bound is suboptimal.
• When using scheme (97) with $\theta = 1$, numerical experiments indicate convergence slows for smaller $h$, approaching $O(h^\alpha)$ for $\alpha \in [1,2)$, indicating potential loss of optimal order.
• The paper establishes discrete versions of Gagliardo-Nirenberg and Sobolev inequalities, which are essential for the error analysis in the $L^p(L^q)$ framework.