An immersed finite-element proposal addresses elliptic interface problems on unfitted meshes, where the interface need not line up with the mesh. It uses nodal, isoparametric IFE spaces on general unfitted quadrilateral and hexahedral meshes and transforms the interface conditions to reference elements—template elements used in the local formulation. The paper reports unisolvent local basis functions for the selected flux-enforcement point, meaning the prescribed nodal information uniquely determines the local basis representation. The authors state that the method also accommodates tensor-valued diffusion coefficients without restrictive angle conditions.
A point chosen inside the reference element
The method’s central geometric step is a rule for choosing where to enforce a discrete flux relation. It identifies a point in the reference element where an interpolated auxiliary gradient is a scalar multiple of the reference normal, with the scalar constrained to the interval [0, 1]. The selected point is then used in constructing the local nodal basis.
The paper identifies centroid-based flux enforcement as a configuration in which the determining quantity may vanish and local unisolvence is lost. The reported counterexamples are a parallelogram with a scalar diffusion coefficient and a square with a tensor-valued coefficient.
The theory comes with conditions
The analysis establishes optimal approximation properties even though flux continuity is not enforced exactly along the full interface. It also establishes optimal error estimates whose constants are independent of the interface’s position relative to the mesh.
The global formulation includes lifting stabilization, and the paper states that the stabilized form is coercive, a mathematical stability property. The approximation analysis explicitly relies on an asymptotically affine mesh condition.
Two refinement examples
The numerical section tests two manufactured-solution examples: one two-dimensional problem and one three-dimensional problem. Each begins with an initial mesh and uses successive uniform refinement. Both examples exhibit second-order convergence in the L2 norm, a measure of overall solution error, and first-order convergence in the broken H1 seminorm, which tracks piecewise gradient error. The reported pattern is consistent with the theoretical analysis.
At the finest listed level for the two-dimensional example, l = 6, the relative L2 error is 2.6125 × 10−5 with a rate of 2.00, while the broken H1 error is 6.0660 × 10−3 with a rate of 1.00. For the three-dimensional example at l = 4, the corresponding relative L2 and broken H1 errors are 5.7263 × 10−4 and 2.6900 × 10−2, with rates of 2.01 and 1.00.
The numerical evidence is limited to these two examples. In those configurations, the calculations are consistent with the theoretical analysis, while the approximation analysis explicitly relies on an asymptotically affine mesh condition.
Taken together, the preprint presents a constructive flux-point rule, reports locally unisolvent basis functions, and establishes optimal approximation and interface-position-independent error estimates under its analysis assumptions. The two refinement examples show the reported second-order L2 and first-order broken H1 rates, but remain numerical examples of the stated construction rather than evidence beyond those assumptions.
Paper data and sources
Original title: Quadrilateral and Hexahedral Immersed Finite Element Methods for Elliptic Interface Problems
Authors: Fangfang Qin, Haifeng Ji
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text