Energy-Equidistributed Moving Mesh PDEs

• Energy-Equidistributed Moving Mesh PDEs are adaptive mesh evolution equations that redistribute nodes to equidistribute prescribed energy densities or meshing functionals.
• They employ variational formulations that enforce equidistribution and alignment conditions, ensuring robust element quality and mesh nonsingularity during simulation.
• EMMPDEs underpin diverse numerical methods in finite element and finite difference schemes, enhancing stability and accuracy in simulations of fluid dynamics, transport, and structure-preserving problems.

Energy-Equidistributed Moving Mesh PDEs (EMMPDEs) are moving-mesh evolution equations in which mesh nodes are redistributed so that a prescribed monitor—often an energy density—or a meshing functional enforcing equidistribution and alignment is balanced across the mesh while element quality is controlled and mesh degeneration is avoided. In the cited literature, EMMPDEs appear in two closely related senses: as adaptive mesh PDEs that equidistribute a physical energy density for conservative or weakly non-conservative systems, and as gradient flows of meshing energies whose stationary points satisfy equidistribution and alignment in a metric tensor. Both viewpoints couple mesh concentration to the evolving structure of the solution or geometry rather than to fixed, a priori node placement (Gao et al., 27 Aug 2025, Huang et al., 2015, Wang et al., 28 Jan 2026).

1. Geometric meaning of equidistribution and alignment

The geometric core of EMMPDEs is the simultaneous enforcement of equidistribution and alignment. In modern moving mesh methods, equidistribution requires mesh cells to have equal metric volume in a prescribed metric tensor $\boldsymbol{M}(\boldsymbol{x})$, while alignment requires element shape and orientation to conform to $\boldsymbol{M}$. For a mapping $$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$ with Jacobian $$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$, the pullback metric is $\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$, and one alignment condition is

$$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$

so that the inverse-pullback tensor

$$\boldsymbol{A}=\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T}$$

has equal eigenvalues at the target mesh state (Wang et al., 28 Jan 2026).

For general $m$-dimensional simplicial meshes in $d$-dimensional space, the same ideas are written in a unifying form. Equidistribution is expressed as

$$|K|_{\mathbb{M}}=\frac{\sigma_h}{N}, \qquad \forall K,$$

with

$\boldsymbol{M}$0

and alignment is expressed as

$\boldsymbol{M}$1

These formulas extend the bulk-mesh viewpoint to curves, surfaces, and domains without changing the underlying equidistribution/alignment logic (zhang et al., 6 Jan 2025).

When EMMPDEs are formulated directly from a monitor function, the same principle appears as equal monitor mass per element. In one dimension,

$\boldsymbol{M}$2

and in conservative PDE settings the monitor may be the energy density $\boldsymbol{M}$3, giving

$\boldsymbol{M}$4

These formulas make explicit that node concentration is increased where $\boldsymbol{M}$5 or $\boldsymbol{M}$6 is large (Toledo-Zucco et al., 2024, Gao et al., 27 Aug 2025).

A recurring ambiguity in the literature is that “energy” does not always denote the same object. In some EMMPDE formulations it means the physical energy density of a PDE, used as a monitor; in others it means a variational meshing energy whose gradient flow generates the mesh motion. The two usages are compatible but not identical.

2. Variational formulations and governing mesh PDEs

A broad variational template for moving mesh generation writes the meshing functional as

$\boldsymbol{M}$7

with $\boldsymbol{M}$8 taken for the inverse coordinate transformation in one standard convention. Classical examples include Generalized Winslow’s functional and Huang’s alignment/equidistribution functional, both combining metric-based size control and shape control through $\boldsymbol{M}$9 (Huang et al., 2015).

A recent unified formulation replaces empirical balancing parameters by an $$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$0-pullback construction. Its functional is

$$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$1

where

$$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$2

and $$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$3. The trace term promotes alignment by penalizing unequal eigenvalues of $$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$4, while the logarithmic determinant term enforces equidistribution of metric volume. The functional is minimized when $$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$5, so equidistribution and alignment are attained simultaneously without empirical parameters (Wang et al., 28 Jan 2026).

Monitor-based EMMPDEs can also be derived directly from a variational principle. In one dimension, equidistribution of energy density is equivalent to minimizing

$$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$6

In multiple dimensions,

$$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$7

with $$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$8. The corresponding location-based EMMPDE is

$$\boldsymbol{x}(\boldsymbol{\xi}): \Omega_c \rightarrow \Omega_p$$9

where $$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$0 is the vector of Euler–Lagrange derivatives, and a velocity-based linearized form is

$$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$1

where $$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$2 is the Jacobian matrix of derivatives of $$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$3. In this formulation the positivity of the monitor $$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$4 is crucial and may be ensured by shifting it with a constant if necessary (Gao et al., 27 Aug 2025).

These formulations show that EMMPDEs are not a single PDE but a family of gradient systems, relaxation systems, or Euler–Lagrange systems whose common target is a mesh satisfying an equidistribution principle together with admissible element shape.

3. Analytical properties: coercivity, convexity, and nonsingularity

The central theoretical question for EMMPDEs is whether the mesh can tangle, collapse, or converge to nonphysical states. A discrete theory based on coercive meshing functionals establishes that, if

$$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$5

and the initial mesh is nonsingular, then the semi-discrete MMPDE mesh remains nonsingular for all $$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$6. Moreover, element altitudes and volumes are bounded below by positive numbers depending only on the number of elements, the metric tensor, and the initial mesh. The discrete meshing functional is monotonically decreasing,

$$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$7

its value converges as $$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$8, and limiting meshes are critical points of the discrete functional. With a stronger condition, the entire mesh trajectory is strongly convergent. The same conclusions extend to fully discrete systems if the time step is sufficiently small and the temporal discretization preserves monotonic energy decrease (Huang et al., 2015).

For the unified $$\boldsymbol{J} = \partial \boldsymbol{x}/\partial \boldsymbol{\xi}$$9-based functional, the analytical picture is sharpened further. The functional is scale invariant, coercive for $\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$0, polyconvex with respect to $\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$1, and geodesically convex with respect to the inverse Jacobian. The cited analysis establishes existence of minimizers and shows that any critical point is a global minimizer; the associated gradient-flow moving mesh PDE has no local minima in the stated geodesic-convex sense, and the minimum element volume is uniformly bounded below for all time (Wang et al., 28 Jan 2026).

For geometric objects embedded in higher-dimensional spaces, nonsingularity must be combined with a kinematic constraint. Surface and curve moving mesh methods therefore project the gradient-flow velocity onto the tangent bundle of the geometric object. For surfaces,

$\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$2

so that motion is tangential and nodes remain on the underlying surface. The analysis shows that if the initial surface mesh is nonsingular, the mesh remains nonsingular for all time and admits limiting meshes that are critical points of the meshing energy (Kolasinski et al., 2019).

A plausible implication is that the stability theory of EMMPDEs is strongest when the mesh motion is derived from a coercive variational principle and the discretization respects the associated energy decay.

4. Discretization and algorithmic realization

The discrete realization of EMMPDEs is typically elementwise and variational. For simplicial meshes, one representative discretization is

$\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$3

with per-element quantities $\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$4 and $\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$5. Mesh motion is then given by the gradient flow

$\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$6

The unified $\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$7-structured formulation yields local assembly formulas in terms of derivatives with respect to $\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$8 and $\boldsymbol{J}^T \boldsymbol{M}\boldsymbol{J}$9, and this supports Jacobian-free quasi-Newton solvers such as DFP or Broyden-type updates coupled to ODE solvers such as BDF. The resulting discrete energy decreases monotonically, and mesh nonsingularity is maintained throughout the evolution (Wang et al., 28 Jan 2026).

In finite element PDE solvers, EMMPDEs are usually embedded in an outer solution–mesh loop. For the regularized long wave equation, the metric tensor is Hessian-based,

$$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$0

with $$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$1 determined by a normalization condition. The mesh is adapted by minimizing an energy functional combining alignment and equidistribution, and the mesh coordinates evolve through a gradient-flow MMPDE. Because the RLW equation contains space-time mixed derivatives, an auxiliary variable is introduced so that the PDE can be rewritten as a coupled system compatible with linear $$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$2 finite elements on moving meshes; time integration is performed with the fifth-order Radau IIA scheme and adaptive time stepping (Lu et al., 2016).

In high-order adaptive finite difference schemes for the shallow water equations, the moving mesh strategy is generated by iteratively solving the Euler–Lagrangian equations of a mesh adaptation functional. The fully discrete PDE–mesh system is advanced by an explicit strong-stability-preserving third-order Runge–Kutta method. In that setting the adaptive mesh is part of a broader construction that is simultaneously well-balanced and energy stable (Zhang et al., 2023).

Across these implementations, the recurring workflow is metric or monitor construction, solution of a mesh PDE or discrete gradient system, transfer or evaluation on the updated mesh, and time advancement of the physical PDE. The exact form of the monitor differs, but the algorithmic skeleton is stable across finite element, finite difference, and neural formulations.

5. Applications and domain-specific extensions

EMMPDEs have been applied to both canonical adaptation tests and fully coupled physical simulations. For the unified variational functional, numerical experiments include function-induced mesh tests, two-dimensional Burgers’ equation with Hessian-based and arc-length-based metrics, and Rayleigh–Taylor instability. The reported outcomes are superior equidistribution, with $$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$3 values closer to 1 than those of classic functionals, robust mesh alignment with comparable interpolation accuracy, stable nonsingularity, monotonic energy descent, and substantial speedup in Newton–Krylov solves, including cases up to $$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$4 faster for large meshes (Wang et al., 28 Jan 2026).

In structure-preserving discretizations of transport problems, moving meshes can be incorporated without losing the governing energy balance. For the boundary controlled transport equation, a scattering-passive structure-preserving finite element method preserves the scattering energy balance at the discrete level, and the one-dimensional moving-mesh extension retains that balance when the time dependence of the basis functions is handled correctly. The moving mesh requires fewer states than the fixed mesh to produce solutions with comparable accuracy and reduces overshoot and oscillations associated with Gibbs phenomenon (Toledo-Zucco et al., 2024).

For the regularized long wave equation, adaptive moving mesh finite elements in one and two dimensions exhibit second-order convergence and concentrate nodes around solitary waves, bore fronts, and other evolving features. With the same number of elements, moving meshes are reported to be an order of magnitude more accurate in $$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$5 and $$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$6 norms than fixed meshes, although the moving-mesh treatment preserves the energy quantity $$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$7 less well because of an extra convection term induced by mesh movement; that error decreases rapidly with refinement (Lu et al., 2016).

The EMMPDE paradigm has also been extended to machine learning. Energy-Equidistributed Moving Sampling Physics-Informed Neural Networks (EEMS-PINNs) use the energy density as the monitor function and couple a solution network with a mesh network. The framework includes both location-based and velocity-based EMMPDEs, and numerical experiments on conservative and non-conservative systems report lower $$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$8 errors, orders-of-magnitude lower relative energy errors than several non-energy-adaptive baselines, stable long-time behavior, and adaptation that often converges in a few rounds (Gao et al., 27 Aug 2025).

Beyond bulk domains, the same equidistribution/alignment principles govern adaptive motion on curves and surfaces. A unifying moving mesh method for curves, surfaces, and domains projects nodal velocities so that mesh points stay on the underlying object, does not require an analytical parametric representation, and is proved nonsingular. Numerical results on circles, ellipses, lemniscates, roses, hyperboloids, cavatappi surfaces, and other examples show concentration in high-curvature regions when curvature-based metrics are used and quasi-uniformity when $$\boldsymbol{J}^{-1}\boldsymbol{M}^{-1}\boldsymbol{J}^{-T} = \theta \boldsymbol{I},$$9 (zhang et al., 6 Jan 2025).

6. Related frameworks, distinctions, and interpretive issues

EMMPDEs are closely related to, but not identical with, several other moving-mesh paradigms. In optimal-transport mesh adaptivity, equidistribution is combined with a minimum-displacement principle, yielding a Monge–Ampère or Monge–Ampère-like equation for a mesh potential. McRae, Cotter, and Budd describe a mixed finite element discretization in which an auxiliary tensor represents the Hessian, and they note that the relaxation or parabolic form of the Monge–Ampère equation corresponds, in the continuum limit, to a gradient-flow-type mesh evolution closely related to EMMPDE behavior (McRae et al., 2016).

A different contrast appears in moving-mesh methods derived from the Onsager variational principle. For the porous medium equation, the mesh does not follow an externally prescribed equidistribution law; instead, mesh and solution co-evolve through minimization of a Rayleighian under mass conservation. The method preserves the energy dissipation structure in semi-discrete and fully discrete forms, naturally captures waiting time phenomena, and differs from classical EMMPDEs in being intrinsic to the physical dissipation mechanism rather than extrinsic to it through a chosen monitor function (Xiao et al., 2024).

The phrase “energy-preserving moving mesh” introduces another distinction. In Hamiltonian or port-Hamiltonian discretizations, the preserved energy is a physical invariant of the PDE rather than a mesh energy. For the BBM equation, rezoning combined with discrete gradient methods yields schemes that exactly preserve a discrete Hamiltonian while adapting the mesh through an equidistribution principle with arc-length-type monitor. For the transport equation, the preserved quantity is a scattering energy balance. These approaches intersect with EMMPDE practice through adaptive node redistribution, but their primary invariant is the PDE’s Hamiltonian or boundary energy flow, not the variational mesh functional (Eidnes et al., 2017, Toledo-Zucco et al., 2024).

A common misconception is therefore that every moving mesh method using an energy-like quantity is an EMMPDE in the same sense. The literature instead separates at least three notions: physical energy preservation, energy-density-based monitor equidistribution, and decay of a meshing functional. These notions can coexist in a single algorithm, but they are analytically distinct.

Another recurring point is portability. In a one-dimensional arbitrary-order ALE moving-boundary algorithm, internal mesh velocities can be replaced by a more traditional mesh movement strategy, including one that seeks to equidistribute an error indicator at each time step, provided the discrete velocity used in the solution update varies linearly within mesh elements. This suggests that EMMPDE components can be inserted into high-order moving-domain frameworks without changing the central solution update mechanism, as long as the mesh-velocity regularity assumed by the discretization is respected (Hubbard et al., 4 Sep 2025).