[Paper Review] Numerical approach to $L_1$-problems with the second order elliptic operators

N. Limić, Mladen Rogina|ArXiv.org|Dec 21, 2007

Differential Equations and Boundary Problems14 references3 citations

TL;DR

This paper presents a numerical method for solving $L^1$-problems involving second-order elliptic operators in divergence form with Dirichlet boundary conditions. It constructs grid-solutions via finite difference schemes with compartmental (M-matrix-like) structure, embeds them into hat functions, and proves strong $L^p$ convergence and weak $\dot{W}^1_p$ convergence for $p \in [1, d/(d-1))$. The key contribution is a provably convergent, monotone scheme preserving physical structure in arbitrary dimensions with rigorous error control.

ABSTRACT

For a second order differential operator $A(\msx) =- abla a(\msx) abla + b'(\msx) abla+ abla \big(\msb''(\msx) \cdot\big)$ on a bounded domain $D$ with the Dirichlet boundary conditions on $\partial D$ there exists the inverse $T(λ, A)= (λI+A)^{-1}$ in $L_1(D)$. If $μ$ is a Radon (probability) measure on Borel algebra of subsets of $D$, then $T(λ, A)μ\in L_p(D), p \in [1, d/(d-1))$. We construct the numerical approximations to $u =T(λ, A)μ$ in two steps. In the first one we construct grid-solutions ${\bf u}_n$ and in the second step we embed grid-solutions into the linear space of hat functions $u(n) \in \dot{W}_p^1(D)$. The strong convergence to the original solutions $u$ is established in $L_p(D)$ and the weak convergence in $\dot{W}_p^1(D)$.

Motivation & Objective

• To develop a stable, monotone numerical scheme for solving $L^1$-problems involving second-order elliptic operators in divergence form.
• To ensure the system matrix of the discrete problem has a compartmental structure (positive diagonal, non-positive off-diagonals, non-negative column sums), mimicking M-matrices.
• To establish strong convergence in $L^p(D)$ and weak convergence in $\dot{W}^1_p(D)$ for $p \in [1, d/(d-1))$.
• To extend existing numerical methods to general domains in $\mathbb{R}^d$ with Lipschitz boundaries and singular data (Radon measures).
• To provide a constructive algorithm for dimension $d=2$ and a general construction for $d \geq 3$ with sufficient conditions on the diffusion tensor.

Proposed method

• Discretize the second-order elliptic operator $A(x)$ in divergence form using finite differences on a structured grid, ensuring consistency and compartmental structure of the system matrix $A_n$.
• Construct grid-solutions $u_n$ as solutions to the discrete linear system $A_n u_n = \mu_n$, where $\mu_n$ is a discrete Radon measure.
• Embed the grid-solutions $u_n$ into the space of continuous piecewise linear (hat) functions $u^{(n)} \in \dot{W}^1_p(D)$ via interpolation.
• Prove strong convergence of $u^{(n)}$ to the true solution $u = T(\lambda, A)\mu$ in $L^p(D)$ and weak convergence in $\dot{W}^1_p(D)$ using compactness and consistency arguments.
• Use a bilinear form $a(v,u)$ and Green's formula to relate the continuous and discrete variational formulations.
• For $d \geq 3$, impose Assumption 4.1 on the diffusion tensor $\{a_{ij}(x)\}$ to guarantee the compartmental structure of $A_n$.

Experimental results

Research questions

• RQ1 Can a monotone finite difference scheme be constructed for second-order elliptic operators in divergence form with general diffusion tensors in arbitrary dimensions?
• RQ2 Does the discrete solution, when embedded into hat functions, converge strongly in $L^p(D)$ and weakly in $\dot{W}^1_p(D)$ for $p \in [1, d/(d-1))$?
• RQ3 What conditions on the diffusion tensor ensure the compartmental structure of the system matrix $A_n$?
• RQ4 How does the proposed scheme perform numerically for problems with singular data (e.g., Dirac measures)?
• RQ5 Can the scheme be implemented efficiently in $d=2$ with a direct algorithmic structure?

Key findings

• For $d=2$, the scheme is algorithmically explicit and can be directly applied, with convergence proven in $W^1_2$ and $W^1_p$ spaces.
• In $d \geq 3$, the compartmental structure of $A_n$ is guaranteed if the tensor-valued function $\hat{a}$ is strictly positive definite, as per Lemma 8.1.
• Numerical Example 7.1 shows $\varepsilon_{1,\text{rel}} = 0.003371$ and $\varepsilon_{\infty,\text{rel}} = 1.6254$ for a problem with a Dirac measure, with the largest error near the singularity.
• Example 7.2 achieves $\varepsilon_{1,\text{rel}} = 8 \times 10^{-6}$ and $\varepsilon_{\infty,\text{rel}} = 4 \times 10^{-3}$ after 210 iterations, confirming high accuracy.
• The method avoids rotations in $d=2$ by using extended stencils, preserving monotonicity and avoiding saw-like artifacts in the solution surface.
• The scheme is robust for singular data (Radon measures), with the inverse operator $T(\lambda, A)$ mapping $\mu \in R(D)$ into $L^p(D)$ and $\dot{W}^1_p(D)$ for $p < d/(d-1)$.