Robust Preconditioning of mixed-dimensional PDEs on 3d-1d domains coupled with Lagrange multipliers

TL;DR

This paper develops a robust preconditioner for mixed-dimensional PDEs modeling micro-circulation, addressing challenges in algebraic structure and parameter robustness, especially concerning the vascular radius.

Contribution

It introduces an operator preconditioning technique tailored for 3d-1d coupled problems with Lagrange multipliers, improving solution efficiency and robustness.

Findings

Preconditioner is robust against most problem parameters.

Effectiveness depends on the vascular radius, which influences well-posedness.

The approach simplifies complex geometries via topological dimensionality reduction.

Abstract

In the context of micro-circulation, the coexistence of two distinct length scales - the vascular radius and the tissue/organ scale - with a substantial difference in magnitude, poses significant challenges. To handle slender inclusions and simplify the geometry involved, a technique called topological dimensionality reduction is employed, which suppresses manifold dimensions associated with the smaller characteristic length. However, the resulting discretized system's algebraic structure presents a challenge in constructing efficient solution algorithms. This chapter addresses this challenge by developing a robust preconditioner for the 3d-1d problem using the operator preconditioning technique. Robustness of the preconditioner is demonstrated with respect to problem parameters, except for the vascular radius. The vascular radius, as demonstrated, plays a fundamental role in mathematical well-posedness of the problem and the preconditioner's effectiveness.