Optimal error bounds for the two point flux approximation finite volume scheme

TL;DR

The paper develops a rigorous error analysis for the TPFA finite volume discretization of the Laplace equation under minimal regularity, i.e., when the exact solution lies in $H^1_0(\Omega)$ and the RHS can belong to $H^{-1}(\Omega)$. It introduces an inflated normal gradient and a weak TPFA formulation to obtain an optimal error bound that splits the discretization error into the sum of an interpolation error $\mathcal{I}_{\mathcal{T}}(\overline{u})$ and a conformity error $\zeta_{\mathcal{T}}(\nabla\overline{u}+\mathbf{F})$, yielding $\frac{1}{2}(\zeta_{\mathcal{T}} + \mathcal{I}_{\mathcal{T}}) \le \delta_{\mathcal{T}} \le 3(\zeta_{\mathcal{T}} + \mathcal{I}_{\mathcal{T}})$. The authors also derive $H^2$-regularity refinements showing first-order convergence in $L^2$ for the solution and a consistent gradient on $d\le 3$ when $\overline{u}\in H^2(\Omega)$. A numerical example with minimal regularity confirms the optimality of the bound, and an appendix extends the results to the transient heat equation via implicit Euler. The work provides robust error control for TPFA schemes on general admissible meshes, informing practical reliability in simulations where regularity cannot be assumed and RHS may be distributional.

Abstract

We consider a finite volume scheme with two-point flux approximation (TPFA) to approximate a Laplace problem when the solution exhibits no more regularity than belonging to $H^1_0(Ω)$. We establish in this case some error bounds for both the solution and the approximation of the gradient component orthogonal to the mesh faces. This estimate is optimal, in the sense that the approximation error has the same order as that of the sum of the interpolation error and a conformity error. A numerical example illustrates the error estimate in the context of a solution with minimal regularity. This result is extended to evolution problems discretized via the implicit Euler scheme in an appendix.

Figures (3)

• Figure 1: Two neighbouring control volumes of an admissible mesh.
• Figure 2: The approximate solution of Problem \ref{['eq:pbellcont']} computed on Mesh1_3.
• Figure 3: Approximation and interpolation errors vs. the size of the mesh.

Theorems & Definitions (35)

• Definition 2.1: Admissible meshes
• Definition 2.2: Discrete derivative and gradients
• Remark 2.3: On the inflated and consistent gradients
• Lemma 2.4: Weak formulation of the scheme
• proof
• Theorem 3.1: Optimal error bound for the approximation of the elliptic problem \ref{['eq:pbellcont']}
• Remark 3.2: On the definition of $\delta_{\mathcal{T}}(\varphi,v)$
• Remark 3.3: On the definition of $\mathcal{G}_{\mathcal{T}}$
• proof : Proof of Theorem \ref{['thm:errest']}
• Remark 3.4: Existence and uniqueness