A control-theoretic framework on moving-mesh methods for solving PDEs

Choosing a good mesh is a central challenge in solving partial differential equations (PDEs), particularly when the solution develops sharp features that move over time. The Moving Mesh PDE (MMPDE) is the standard 𝑟-adaptive method that relocates a fixed set of mesh nodes to track these features, yet it has been studied almost exclusively through numerical analysis. We show that the MMPDE is, in control-theoretic terms, a proportional (P) feedback controller. This reading reframes mesh adaptation as a control-design problem, and it reveals an inherent limitation of the MMPDE. For PDEs whose solutions develop moving features, such as travelling shocks, fronts, and vortices, a proportional controller cannot keep the mesh matched to the feature. To resolve this, we propose mesh controllers, procedural moving-mesh strategies in which (i) the controller structure is set by the Internal Model Principle from the PDE feature motion, (ii) the controller gains are set adaptively when the feature motion varies in time, and (iii) a feedforward term is added when the feature’s transport velocity is known from prior physical knowledge of the problem. We develop two instances of mesh controllers, a PI-MMPDE and an Adaptive-PI MMPDE, and test them on two benchmark PDEs: the 2D Burgers and LeVeque flows, where the mesh controllers show substantial improvement over the conventional MMPDE baselines.