A Unified Variational Functional for Equidistribution and Alignment in Moving Mesh Adaptation

TL;DR

This paper introduces a new variational functional for moving mesh adaptation that balances mesh size and anisotropy without empirical parameters, ensuring robust and efficient mesh generation.

Contribution

A novel variational functional based on an A-pullback formulation that enforces equidistribution and alignment with proven mathematical properties and an efficient discretization.

Findings

Functional achieves balanced control of mesh size and anisotropy.

Theoretical properties such as coercivity and convexity are established.

Numerical experiments demonstrate robustness and efficiency.

Abstract

Existing variational mesh functionals often suffer from strong nonlinearity or dependence on empirical parameters.We propose a new variational functional for adaptive moving mesh generation that enforces equidistribution and alignment through an $\boldsymbol A$-pullback formulation, where $\boldsymbol A=\boldsymbol J^{-1}\boldsymbol M^{-1}\boldsymbol J^{-T}$. The functional combines a trace-based term with a logarithmic determinant term, achieving balanced control of mesh size and anisotropy without empirical parameters. We establish coercivity, polyconvexity, existence of minimizers, and geodesic convexity with respect to the inverse Jacobian, and derive a simplified geometric discretization leading to an efficient moving mesh algorithm. Numerical experiments confirm the theoretical properties and demonstrate robust adaptive behavior for function-induced meshes and Rayleigh-Taylor instability simulations.