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The aggregated unfitted finite element method for elliptic problems
Santiago Badia, Francesc Verdugo, Alberto F. Martín
TL;DR
Unfitted finite element methods avoid difficult body-fitted meshing but can become severely ill conditioned because of small cut cells. This paper constructs conforming aggregated finite element spaces using cell-based constraints, and proves and numerically confirms body-fitted-like conditioning and optimal convergence for elliptic problems.
Problem
Unfitted finite element methods can have condition numbers controlled by arbitrarily small cut cells, obstructing efficient iterative solution of large linear systems.
Method
The aggregated unfitted finite element method removes potentially problematic degrees of freedom from conforming Lagrangian spaces using constraints derived from cell aggregates.
Results
The method yields condition numbers independent of small cut cells, scales as h^-2 like standard body-fitted finite elements, and recovers optimal finite element convergence order.
Takeaways & Limitations
The aggregated spaces provide an easy-to-implement formulation for continuous and discontinuous methods, supported by 2D and 3D experiments with first- and second-order interpolations.
Takeaways & Limitations
The analysis assumes fixed geometric and aggregation parameters, including bounded aggregate distance and quasi-uniform background meshes, while numerical setup uses geometry perturbation tolerances for non-aggregated spaces.
Abstract
from arXiv · showhide
Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hinders the practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the new method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.
1. Introduction
Unfitted finite element methods simplify meshing complex or evolving geometries but suffer from conditioning problems linked to arbitrarily small cut cells. The paper develops an aggregated formulation intended to remove these difficulties while remaining broadly usable in finite element codes.
- Unfitted finite element methods are useful when body-fitted mesh generation is difficult, including moving-interface, optimization, medical, and large-scale simulation settings.
- Ill conditioning depends on both background element size and arbitrarily small cut-cell size, limiting scalable iterative solvers for large systems.
- The work targets condition numbers governed only by background mesh size, as in standard body-fitted finite elements.
- The aggregated formulation is designed for continuous and discontinuous formulations, high-order interpolation, and implementation in an existing parallel finite element code.
- The method removes potentially problematic degrees of freedom through constraints derived from cell aggregates without altering conformity of the original finite element space.
- The article analyzes the formulation for elliptic problems and evaluates it through numerical experiments after introducing the embedded-boundary setup and aggregate-based spaces.
2. Embedded boundary setup and cell aggregation
The embedded-boundary setup uses a simple artificial background mesh whose cells are classified relative to the physical domain. Cell aggregation repeatedly joins cut cells to connected interior-cell aggregates, producing bounded-size aggregates under a path-length assumption.
- Embedded boundary setup: The physical domain lies inside a simply meshed artificial domain, with cells classified as interior, external, or cut according to their intersection with the physical domain.
- Embedded boundary setup: The background mesh is assumed quasi-uniform, with Cartesian hexahedra in 3D or quadrilaterals in 2D considered as the main setting.
- Cell aggregation scheme: Each aggregate contains one interior root cell, while every cut cell belongs to exactly one aggregate; alternative schemes can also include sufficiently large cut cells.
- Cell aggregation scheme: Algorithm 2.1 marks interior cells as touched, then repeatedly aggregates untouched cells to connected touched cells until all cells are assigned.
- Cell aggregation scheme: The aggregation procedure merges neighboring cells by removing internal boundaries, and it extends to arbitrary spatial dimensions, including complex 3D domains.
- Aggregate-size bound: If every cut cell connects to an interior cell through a path of at most γmax cells, aggregate size is bounded by (2γmax + 1)hmax.
- Aggregate-size bound: Under the stated corner-sharing assumption, γmax equals 2 in 2D and 3 in 3D, giving aggregate-size bounds of 5hmax and 7hmax respectively.
3. Aggregated unfitted Lagrangian finite element spaces
The aggregated unfitted space modifies the standard active Lagrangian finite element space by constraining problematic outer-node degrees of freedom through cell aggregates. This preserves conformity and supports implementation in existing finite element codes while avoiding small-cut-cell ill conditioning.
- The construction uses conforming Lagrangian finite elements with polynomial spaces Pq on simplices or Qq on cubes and a uniform polynomial order across cells.The nodal Lagrangian basis provides the node-to-shape-function correspondence used to define the constraints and extensions.
- The active space is standard in unfitted methods but can produce arbitrarily ill-conditioned systems when the weak form is integrated only over the physical domain.
- The aggregated space is defined by interior nodal values and covers the physical domain without the ill-conditioning problems of the active space.The interior space alone avoids the conditioning problem but is not defined over the complete physical domain; aggregation combines coverage with constrained extension.
- The aggregated space is constructed from the active space by adding constraints for outer nodes identified through a map to interior cells.Outer nodes are associated with interior cells through the cell aggregates, and their values are obtained by extrapolation from those cells.
- The resulting space retains conformity while eliminating conflictive outer nodes through cell-wise local constraints.These constraints can be applied during cell-wise assembly, making implementation straightforward even for nonadaptive codes without non-conforming-mesh support.
- An alternative aggregate-based space is equivalent in some respects but is harder to implement with standard inter-element continuity, whereas the proposed space requires minimal code changes.The two approaches coincide for discontinuous Galerkin formulations, where all degrees of freedom belong to cells.
4. Approximation of elliptic problems
The method formulates elliptic problems on unfitted, H1-conforming finite element spaces and imposes Dirichlet conditions weakly with Nitsche’s method. Aggregation removes the conditioning problems associated with cut cells while retaining a practical Galerkin formulation.
- Problem and finite element spaces: The model problem is Poisson’s equation with Dirichlet and Neumann boundary conditions on a polygonal domain.The formulation is stated for constant diffusion, although the ideas apply to broader elliptic problems with H1-stability.
- Problem and finite element spaces: The background mesh is conforming but need not align with the physical boundary, so cells may be cut by the domain boundary.The analysis considers both the usual active-cell space and the aggregated space for comparison.
- Variational formulation: Dirichlet boundary conditions are imposed weakly using Nitsche’s method rather than being included strongly in the unfitted approximation space.The element bilinear form contains the integration-by-parts terms and the additional Nitsche contribution.
- Variational formulation: The aggregated formulation uses an enhanced space and constructs the global problem from element contributions over the active mesh.The global operator and right-hand side are assembled as sums of element-level terms, and the resulting problem is expressed through the extension operator.
- Conditioning and coercivity: For standard unfitted spaces, cut cells can make the coercivity parameter τK impossible to bound a priori, whereas aggregation restores the body-fitted scaling τK = 100/hK used in the experiments.The aggregated space makes τK bounded in terms of element size again.
- Conditioning and coercivity: The usual active-cell space can produce arbitrarily high condition numbers because the interface may leave cells with arbitrarily small physical intersections.Using the aggregated space instead leads to the stated condition-number result in Corollary 5.9.
5. Numerical analysis
The numerical analysis establishes stability of the aggregated extension and derives well-posedness, conditioning, and approximation estimates under quasi-uniformity and fixed aggregation assumptions. The key result is a stiffness-matrix condition number bounded by κ(A) ≲ h^-2.
- Analytical assumptions: The analysis assumes a quasi-uniform background mesh and constants independent of both mesh size and cut location.Constants may depend on the domain, boundary, finite element order, and fixed maximum aggregation distance.
- Extension stability: The extension operator maps interior degrees of freedom to constrained outer values through cell-wise and global constraint matrices.The cell-wise operator has the form E_AK u = [u_K, C-hat_AK u_K], while the global construction uses C.
- Extension stability: The extension-matrix bounds are independent of cut location and aggregate size, supporting norm equivalence for functions in the aggregated space.The resulting bounds underpin continuity of the extension operator and conditioning estimates for the aggregated mass matrix.
- Mass-matrix conditioning: The aggregated mass matrix has a bounded condition number derived from upper and lower norm estimates for the extended finite element functions.The analysis obtains equivalence between the physical L2 norm and the Euclidean norm of interior degrees of freedom.
- Well-posedness: The analysis establishes an extended inverse inequality and a trace inequality needed to prove coercivity of the aggregated unfitted bilinear form.These estimates use standard inverse inequalities together with stability of the extension and bounded aggregate neighborhoods.
- Well-posedness: For sufficiently large βK, the bilinear form is coercive and continuous, so the aggregated unfitted problem has exactly one solution.The required lower bound for βK is independent of mesh size and the intersection of the interface with the active mesh.
- Condition number: κ(A) ≲ h^-2 for the aggregated stiffness matrix, matching the standard finite-element scaling for body-fitted meshes.The bound is stated in Corollary 5.9 for the linear system matrix arising from the aggregated formulation.
- Error estimates: The aggregated unfitted problem satisfies an a priori error estimate under the stated smoothing and regularity assumptions.The estimate is introduced in Theorem 5.11 for solutions with the required Sobolev regularity.
6. Numerical experiments
The experiments show that cell aggregation stabilizes unfitted finite element systems across domain positions and mesh refinements while preserving convergence behavior. Standard unfitted spaces become highly sensitive to geometry, interpolation order, and dimension, whereas aggregated spaces achieve body-fitted-like conditioning.
- Moving domain experiment: Aggregation slightly increases energy-norm error on coarse moving-domain meshes, but the increment becomes negligible on finer meshes.The coarse-mesh increase is attributed to the moving-domain experiment's resolution.
- Convergence test: Aggregate sizes scale as 2h in 2D and tend to 3h in 3D under mesh refinement.These observations agree with the theoretical aggregate-size bounds.
- Convergence test: Aggregated stiffness-matrix condition numbers scale as h^-2, matching standard body-fitted finite elements; unaggregated condition numbers behave erratically.The standard formulation remains sensitive to cut-cell configurations, interpolation order, and spatial dimension.
- Convergence test: Without aggregation, the finest 3D meshes with second-order interpolation can be unsolvable because condition numbers reach order 10^30.The incomplete error curve reflects failure to solve those systems with standard double-precision sparse direct solvers.
7. Conclusions
The paper proposes cell-aggregate-based finite element spaces for unfitted elliptic problems and establishes their conditioning and convergence properties. Numerical experiments in 2D and 3D confirm the theoretical results for first- and second-order interpolations.
- Method: The method constructs enhanced finite element spaces by merging cut cells with interior cells and imposing cell-wise constraints.It is designed for both continuous and discontinuous formulations and is implementable in existing finite element codes.
- Theory: The analysis proves conditioning independent of small cut cells and matrix scaling with the inverse square of the background-mesh size, as in standard finite elements.The result is stated for elliptic problems.
- Theory: The method recovers optimal finite element convergence order and bounds Nitsche's penalty parameter from above.These properties are part of the paper's mathematical analysis.
- Numerical validation: Two- and three-dimensional experiments with first- and second-order interpolations confirm the theoretical conditioning and convergence results.The numerical tests cover both dimensions and both interpolation orders.