A PDAE formulation of parabolic problems with dynamic boundary conditions

TL;DR

This paper introduces a novel PDAE formulation for parabolic problems with dynamic boundary conditions, explicitly including boundary constraints via Lagrange multipliers, and proves its well-posedness.

Contribution

It presents the first PDAE-based formulation for such problems, explicitly incorporating boundary constraints with Lagrange multipliers and establishing well-posedness.

Findings

The formulation is mathematically well-posed.

Boundary conditions are effectively incorporated as algebraic constraints.

The approach provides a new framework for analyzing parabolic PDEs with dynamic boundaries.

Abstract

The weak formulation of parabolic problems with dynamic boundary conditions is rewritten in form of a partial differential-algebraic equation. More precisely, we consider two dynamic equations with a coupling condition on the boundary. This constraint is included explicitly as an additional equation and incorporated with the help of a Lagrange multiplier. Well-posedness of the formulation is shown.