[Paper Review] On discrete functional inequalities for some finite volume schemes

Marianne Bessemoulin‐Chatard, Claire Chainais-Hillairet+1 more|arXiv (Cornell University)|Feb 22, 2012

Advanced Numerical Methods in Computational Mathematics19 references4 citations

TL;DR

This paper establishes discrete Gagliardo-Nirenberg-Sobolev and Poincaré-Sobolev inequalities for finite volume schemes on general meshes with arbitrary boundary conditions, leveraging the continuous embedding of BV(Ω) into L^{N/(N-1)}(Ω). The key contribution is a robust framework for convergence analysis of nonlinear and anisotropic elliptic and parabolic problems using discrete functional inequalities in the DDFV (discrete duality finite volume) context.

ABSTRACT

We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincaré-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the space $BV(Ω)$ into $L^{N/(N-1)}(Ω)$ for a Lipschitz domain $ Ω\subset \mathbb{R}^{N}$, with $N \geq 2$. Finally, we give several applications to discrete duality finite volume (DDFV) schemes which are used for the approximation of nonlinear and non isotropic elliptic and parabolic problems.

Motivation & Objective

• To derive discrete functional inequalities for finite volume schemes applicable to nonlinear and anisotropic elliptic and parabolic problems.
• To extend Poincaré-Sobolev and Gagliardo-Nirenberg-Sobolev inequalities to general finite volume meshes with arbitrary boundary values.
• To provide a theoretical foundation for convergence and error analysis of DDFV schemes using BV space embeddings.
• To unify and generalize existing discrete inequalities under a single analytical framework based on continuous embeddings.
• To support the convergence analysis of DDFV schemes for problems with Dirichlet, Neumann, or mixed boundary conditions.

Proposed method

• Utilizes the continuous embedding of the space BV(Ω) into L^{N/(N-1)}(Ω) for Lipschitz domains Ω ⊂ ℝ^N, N ≥ 2, as the foundational analytical tool.
• Applies this embedding to derive discrete inequalities for finite volume approximations, particularly in the DDFV (discrete duality finite volume) framework.
• Employs a discrete gradient formulation based on cell and dual-cell unknowns, with interface fluxes defined over mesh interfaces.
• Uses Cauchy-Schwarz and geometric estimates involving the angle α_T of the mesh to bound discrete seminorms in terms of the L^q norm.
• Derives inequalities for both homogeneous Dirichlet and Neumann boundary conditions by adapting the underlying continuous embedding argument.
• Establishes bounds on the L^q norm of discrete functions in terms of discrete W^{1,p} seminorms, with constants depending on domain geometry and mesh regularity.

Experimental results

Research questions

• RQ1 Can discrete Gagliardo-Nirenberg-Sobolev and Poincaré-Sobolev inequalities be established for finite volume schemes with arbitrary boundary conditions on general meshes?
• RQ2 How can the continuous embedding of BV(Ω) into L^{N/(N-1)}(Ω) be adapted to derive robust discrete functional inequalities in the finite volume context?
• RQ3 What are the dependence and stability of the discrete inequality constants on mesh regularity, particularly the angle α_T?
• RQ4 How can these inequalities be extended to DDFV schemes with mixed or non-homogeneous boundary conditions?
• RQ5 Can these inequalities support convergence analysis for nonlinear and anisotropic elliptic and parabolic problems in the DDFV framework?

Key findings

• A discrete Poincaré-Sobolev inequality is established for DDFV schemes with homogeneous Dirichlet boundary conditions in 2D, valid for 1 ≤ p < 2 and 1 ≤ q ≤ p* = 2p/(2-p), and for p ≥ 2 and 1 ≤ q < ∞.
• The constant in the inequality depends on p, q, the boundary part ̺0, the domain Ω, and the mesh's minimal angle α_T, with a factor of 1/(sin(α_T))^{1/p} in the denominator.
• For the case p=2, the inequality reduces to a discrete Poincaré-type bound with constant proportional to 1/sin(α_T), confirming robustness under mesh distortion.
• The method enables convergence analysis for nonlinear and anisotropic PDEs by providing uniform bounds on discrete solutions in L^q norms.
• The approach generalizes previous results by avoiding restrictive mesh assumptions (e.g., orthogonality) and extending to arbitrary boundary values.
• The framework supports both Dirichlet and Neumann boundary conditions through a unified embedding-based argument, enhancing applicability to real-world problems.