A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

TL;DR

This paper introduces a novel PDE-constrained optimization approach for solving 3D-1D coupled problems with discontinuous interface solutions, enhancing robustness and flexibility in complex geometries.

Contribution

It extends previous methods to handle discontinuous solutions at interfaces using domain decomposition and independent discretizations, enabling efficient parallel computation.

Findings

Method accurately solves problems with known analytical solutions.

Robustness against geometrical complexity and mesh size variations.

Suitable for practical applications with complex configurations.

Abstract

A numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined function spaces a well posed 3D-1D problem is obtained from the original fully 3D problem and the solution is then found by a PDE-constrained optimization reformulation. This is a domain decomposition strategy in which unknown interface variables are introduced and a suitably defined cost functional, expressing the error in fulfilling interface conditions, is minimized constrained by the constitutive equations on the subdomains. The resulting discrete problem is robust with respect to geometrical complexity thanks to the use of independent discretizations on the various subdomains. Meshes of different sizes can be used without affecting the conditioning of the discrete linear system, and this is a peculiar aspect of the considered formulation. An efficient resolution strategy is further proposed, based on the use of a gradient based solver and yielding a method ready for parallel implementation. A numerical experiment on a problem with known analytical solution shows the accuracy of the method, and two examples on more complex configurations are proposed to address the applicability of the approach to practical problems.