[Paper Review] Maximal $\bf L^p$ analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra

Buyang Li, Weiwei Sun|arXiv (Cornell University)|Jan 29, 2015

Advanced Mathematical Modeling in Engineering27 references3 citations

TL;DR

This paper establishes optimal $ L^p $ error estimates and discrete maximal $ L^p $ regularity for Galerkin finite element solutions of parabolic equations with nonsmooth coefficients in convex polyhedral domains. By proving the analyticity of the discrete semigroup and leveraging weighted $ L^p $ estimates under $ W^{1,N+eta} $-regularity of the diffusion coefficient, it achieves optimal convergence rates without logarithmic factors, extending maximal $ L^p $ theory to non-smooth coefficients.

ABSTRACT

The paper is concerned with Galerkin finite element solutions for parabolic equations in a convex polygon or polyhehron with a diffusion coefficient in $W^{1,N+ε}$ for some $ε>0$, where $N$ denotes the dimension of the domain. We prove the analyticity of the semigroup generated by the discrete elliptic operator, the discrete maximal $L^p$ regularity and the optimal $L^p$ error estimate of the finite element solution for the parabolic equation.

Motivation & Objective

• To establish discrete maximal $ L^p $ regularity for finite element solutions of parabolic equations with nonsmooth coefficients.
• To prove optimal $ L^p $ error estimates for Galerkin finite element methods in convex polyhedral domains.
• To extend maximal $ L^p $ analysis to cases where the diffusion coefficient lies in $ W^{1,N+eta} $ for $ \beta > 0 $, relaxing the smoothness assumptions.
• To remove logarithmic factors in error estimates by leveraging $ W^{1,N+\beta} $-regularity of the coefficient and domain convexity.
• To provide a unified framework for $ L^p $-norm analysis of finite element solutions under minimal regularity assumptions on coefficients.

Proposed method

• Prove analyticity of the discrete semigroup $ \{E_h(t)\}_{t>0} $ generated by the finite element operator $ A_h $, ensuring stability in $ L^p $-norms.
• Establish a discrete resolvent estimate $ \| (\lambda + A_h)^{-1} v_h \|_{L^\infty} \leq C \lambda^{-1} \| v_h \|_{L^\infty} $ for $ \lambda \in \Sigma_{\varphi + \pi/2} $, crucial for maximal regularity.
• Use weighted $ L^p $-estimates and Riesz transform boundedness on $ L^q $ to derive $ \| \nabla u_h \|_{L^p((0,T);L^q)} \leq C \| f \|_{L^p((0,T);L^q)} $, the core of maximal $ L^p $ regularity.
• Apply the Riesz transform boundedness criterion via local $ L^q $-norm control of $ \nabla u $, relying on the De Giorgi-Nash-Moser theory and $ W^{1,N+\beta} $-regularity.
• Use the $ W^{1,s} \hookrightarrow L^q $ and $ L^s \hookrightarrow W^{-1,q} $ embeddings with $ s = qN/(q+N) $ to control lower-order terms in energy estimates.
• Employ a cut-off function technique and iterative scaling to derive the local $ L^q $-norm estimate (A.10), which implies boundedness of the Riesz transform and hence maximal regularity.

Experimental results

Research questions

• RQ1 Can discrete maximal $ L^p $ regularity be established for parabolic finite element schemes with nonsmooth coefficients in convex polyhedral domains?
• RQ2 Does the $ L^p $ error estimate for finite element solutions remain optimal when the diffusion coefficient is only in $ W^{1,N+\beta} $, not $ C^2 $?
• RQ3 Can the logarithmic factor in previous error estimates be removed under $ W^{1,N+\beta} $-regularity of the coefficient and convex domain geometry?
• RQ4 Is the Riesz transform $ \nabla A^{-1/2} $ bounded on $ L^q(\Omega) $ under $ W^{1,N+\beta} $-regularity of the coefficient and convex polyhedral domain?
• RQ5 What is the sharp regularity condition on the coefficient $ a $ that ensures optimal $ L^p $ convergence for finite element solutions of parabolic equations?

Key findings

• The discrete semigroup $ \{E_h(t)\}_{t>0} $ generated by the finite element operator $ A_h $ is analytic on $ L^\infty \cap S_h $, with uniform bounds independent of the mesh size.
• The discrete maximal $ L^p $ regularity estimate holds: $ \| \partial_t u_h \|_{L^p((0,T);L^q)} + \| A_h u_h \|_{L^p((0,T);L^q)} \leq C_{p,q} \| f \|_{L^p((0,T);L^q)} $ for all $ 1 < p,q < \infty $, with no logarithmic factor.
• Optimal $ L^p $ error estimates are obtained: $ \| P_h u - u_h \|_{L^p((0,T);L^q)} \leq C_{p,q} ( \| P_h u^0 - u_h^0 \|_{L^q} + \| P_h u - R_h u \|_{L^p((0,T);L^q)} ) $, under $ W^{1,N+\beta} $-regularity of $ a $.
• The Riesz transform $ \nabla A^{-1/2} $ is bounded on $ L^q(\Omega) $ for $ 1 < q < \infty $, provided the coefficient $ a \in W^{1,N+\beta}(\Omega) $ and $ \Omega $ is convex, which is essential for maximal regularity.
• The local $ L^q $-norm control (A.10) for solutions of $ A u = 0 $ is established via iterative scaling and $ W^{1,s} \hookrightarrow L^q $ embeddings, confirming the Riesz transform boundedness.
• The results hold for $ N = 2,3 $, and the method extends to general convex polyhedral domains under the stated coefficient regularity, removing prior smoothness restrictions.