Residual Velocities in Steady Free Boundary Value Problems of Vector Laplacian Type

TL;DR

This paper develops a technique to analyze the well-posedness of vector Laplacian free boundary problems with interface conditions, and demonstrates how residual velocities can improve numerical solutions in two-phase flow models.

Contribution

It introduces a method to determine well-posedness of vector elliptic interface problems and explores residual velocities for efficient numerical solutions.

Findings

Well-posedness criteria are clearly established for certain interface conditions.

Residual velocities can be chosen to enhance numerical stability and convergence.

Numerical example demonstrates advantages of non-physical residual velocities in two-phase flow simulation.

Abstract

This paper describes a technique to determine the linear well-posedness of a general class of vector elliptic problems that include a steady interface, to be determined as part of the problem, that separates two subdomains. The interface satisfies mixed Dirichlet and Neumann conditions. We consider ``2+2'' models, meaning two independent variables respectively on each subdomain. The governing equations are taken to be vector Laplacian, to be able to make analytic progress. The interface conditions can be classified into four large categories, and we concentrate on the one with most physical interest. The well-posedness criteria in this case are particularly clear. In many physical cases, the movement of the interface in time-dependent situations can be reduced to a normal motion proportional to the residual in one of the steady state interface conditions (the elliptic interior problems and the other interface conditions are satisfied at each time). If only the steady state is of interest, one can consider using other residuals for the normal velocity. Our analysis can be extended to give insight into choosing residual velocities that have superior numerical properties. Hence, in the second part, we discuss an iterative method to solve free boundary problems. The advantages of the correctly chosen, non-physical residual velocities are demonstrated in a numerical example, based on a simplified model of two-phase flow with phase change in porous media.