Unfitted Multi-Level hp Refinement for Localized and Moving Solution Features

2026-04-28

AI Summary

This work addresses the high computational cost of traditional adaptive finite element methods when solving partial differential equations featuring singularities, steep gradients, discontinuities, or moving sources, which often require frequent mesh reconfiguration and handling of non-conforming interfaces. The authors propose a non-conforming multilevel hp-refinement approach that superimposes local overlay meshes—freely insertable, translatable, or removable—onto a fixed background grid. The global solution is constructed by summing approximation spaces across all levels, with homogeneous constraints enforced on artificial overlay boundaries to ensure C⁰ continuity. This method eliminates the need for mesh alignment, substantially reduces inter-level coupling complexity, reuses standard element routines, and maintains exponential convergence for problems with discontinuities, singularities, or moving sources. It achieves accuracy comparable to conforming methods with fewer degrees of freedom, yielding a superior error-to-cost ratio.

Abstract

Localized features such as singularities, sharp gradients, discontinuities, and moving sources require adaptive finite element discretizations. Conventional refinement strategies introduce significant computational overhead through mesh-topology modifications, constraint handling for non-matching interfaces, and repeated remeshing with state transfer. This work presents an unfitted multi-level hp-refinement strategy that enriches a fixed base discretization by independently positioned overlay meshes. The global approximation space is constructed by superposition of the active spaces across all refinement levels, while homogeneous constraints on artificial overlay boundaries ensure global $C^0$ continuity. Coupling between non-matching meshes is assembled over admissible integration regions defined by intersections of element partitions, enabling reuse of standard element-level finite element routines within a lightweight superposition framework. In contrast to fitted multi-level approaches, overlay boundaries are not required to align with underlying mesh interfaces. This reduces inter-level coupling and allows refinement zones to be inserted, translated, and removed without modifying the base discretization. Numerical studies for discontinuous and singular benchmark problems, as well as a moving source, demonstrate the performance of the method. The unfitted approach retains exponential convergence for non-smooth problems and achieves improved error-to-cost ratios compared to fitted multi-level hp-refinement. For representative cases, comparable accuracy is obtained with substantially fewer degrees of freedom, while localized high-order refinement accurately tracks moving features.

Problem

• localized features
• moving sources
• adaptive finite element
• mesh refinement
• non-matching interfaces

Innovation

• unfitted refinement
• multi-level hp-adaptivity
• overlay meshes
• superposition FEM
• moving features