Daniel Fuster

Le 21 mai 2026

Title: Regularization models, errors and its applications

The use of regularization methods to represent discontinuities and jumps in the solution is appealing for the development of multiphase and multiscalenumerical methods, as it allows techniques originally developed for single-phase flows to be easily extended to situations in which variables and their derivatives are not necessarily continuous. The one-fluid method, inwhich discontinuities in material properties and solution jumps are smearedover a thin layer of controlled thickness, is a representative example. These approaches have become very popular in both scientific and industrial communities due to their simplicity, robustness, and their ability to converge to the exact solution of the discontinuous problem. However, they are known to suffer from reduced accuracy, particularly near interfaces or discontinuities, where derivatives are often contaminated by numerical artifacts resulting from the regularization process.

In this presentation, we introduce a new approach for the arbitrary-order estimation of the errors introduced by regularized methods.  We show that, for  linear elliptic problems, solutions of regularized formulations typically converge only at first order to the exact discontinuous solution. Remarkably, this apparent limitation can be overcome through a posteriori sharp solution reconstruction [Fuster et al, JCP, 2024,2025]. By combining a multiscale framework with asymptotic analysis, the solution of the regularized problem can  be corrected to accurately represent discontinuous solutions, including jumps in primitive variables and their fluxes, without modifying the numerical method used to compute the original regularized solution.  In this presentation, in addition to the solution of simplified elliptic problems, the impact of regularization errors on the development of interfacial instabilities will be discussed.

This approach opens the door to a new class of numerical methods that provide a dual regularized/sharp representation of interfaces. In addition, it offers a natural framework for incorporating subgrid physical models of phenomena occurring along lines, interfaces, or thin regions at controlled accuracy.