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An unfitted $hp$-interface penalty finite element method for elliptic interface problems
Haijun Wu, Yuanming Xiao
TL;DR
Elliptic interface problems on unfitted meshes call for high-order finite element methods without requiring the mesh to fit the interface. This paper proposes hp-IPFEM in two and three dimensions, with symmetric and non-symmetric formulations; its broken H1 estimates are optimal in h and suboptimal in p by half an order.
Problem
Classical finite element methods require meshes fitted to interfaces and boundaries to achieve optimal convergence, motivating high-order methods on simpler unfitted meshes.
Source: §1 · pp. 1, 3Method
Results
Broken H1 error estimates are h-optimal and p-suboptimal by half an order in two and three dimensions; L2 error estimates are also derived by duality.
Source: §1 · p. 3Takeaways & Limitations
The analysis establishes hp-IPFEM error estimates for elliptic interface problems on unfitted meshes in both two and three dimensions.
Source: §1 · pp. 3–4The estimate constants may depend on the coefficient jump across the interface, and deriving explicit dependence is left for future work.
Source: §1 · p. 4
Abstract
from arXiv · showhide
An $hp$ version of interface penalty finite element method ($hp$-IPFEM) is proposed for elliptic interface problems in two and three dimensions on unfitted meshes. Error estimates in broken $H^1$ norm, which are optimal with respect to $h$ and suboptimal with respect to $p$ by half an order of $p$, are derived. Both symmetric and non-symmetric IPFEM are considered. Error estimates in $L^2$ norm are proved by the duality argument.
1 Introduction
The paper develops an hp interface penalty finite element method for elliptic interface problems on unfitted meshes, addressing the challenge of high-order approximation without interface-fitted grids. Its analysis covers two and three dimensions and both symmetric and non-symmetric methods.
1 Introduction
- Unfitted meshes avoid repeated remeshing for moving interfaces and can sidestep inverted elements near curved interfaces during local refinement.The passage identifies these as motivations for unfitted methods, particularly in three dimensions for the inverted-element issue.Source: §1 · p. 2
- Earlier unfitted finite element approaches include basis-modification methods, penalty methods, and weighted-flux methods, with reported accuracy and convergence properties varying by method.The cited discussion contrasts linear immersed and multiscale approaches, Babuška’s penalty method, Hansbo and Hansbo’s method, and Massjung’s hp-DG method.Source: §1 · p. 3
- The proposed hp-IPFEM penalizes jumps in solution value and flux and uses arithmetic average flux; its broken H1 error estimates are h-optimal and p-suboptimal by half an order in two and three dimensions.The paper also derives L2 error estimates by duality for symmetric and non-symmetric methods.Source: §1 · pp. 3–4
- The analysis bounds the arithmetic average flux through one component and the flux jump, then applies a local inverse trace inequality on one side of the interface element.This avoids proving the inequality on both subelements and avoids handling complicated interface positions, including cases where one subelement may not be shape regular.Source: §1 · p. 4
- The constants in the paper’s estimates may depend on the coefficient jump across the interface; explicit dependence on that jump is left for future work.The authors say an explicit treatment could complicate the analysis and obscure the basic idea.Source: §1 · p. 4
2 Formulation of hp-interface penalty finite element methods
The formulation defines an hp finite element approximation on an unfitted mesh and an interface bilinear form with penalties on solution and flux jumps. The method is considered in symmetric and non-symmetric forms.
2 Formulation of hp-interface penalty finite element methods
- The mesh is a conforming, quasi-uniform, regular partition, and the interface induces a partition into segments or patches associated with mesh elements.Elements have diameters comparable to the global mesh size.Source: §2 · p. 5
- Polynomial degree p defines the elementwise approximation space, with total degree on simplices and separate degree limits on parallelograms and parallelepipeds.The interface penalty space is distinguished from the hp-continuous finite element space and is not contained in H1(Ω).Source: §2 · p. 5
- The discrete method seeks uh in the interface penalty approximation space so that the bilinear form ah(uh, vh) equals Fh(vh) for every test function vh in that space.The continuous solution satisfies the corresponding variational formulation over V.Source: §2 · p. 6
- For β=1 the bilinear form is symmetric, while β=-1 gives the non-symmetric method; the paper presents these two cases.The formulation also includes nonnegative penalty parameters, with their conditions specified later.Source: §2 · p. 6
- The cited Hansbo–Hansbo method omits the J1 penalty and uses weighted averages; its linear two-dimensional case has optimal H1 and L2 error estimates.The cited Massjung hp-unfitted DG method penalizes solution and flux jumps and has h-optimal but p-suboptimal broken H1 convergence in two dimensions.Source: §2 · pp. 6–7
3 Some preliminary lemmas
The preliminary results provide local trace estimates and hp approximation tools used in the later error analysis. They also invoke Sobolev extension operators for functions defined on the subdomains.
3 Some preliminary lemmas
- For sufficiently small mesh size, Lemma 3.1 gives trace and inverse trace inequalities on either side of each interface segment or patch.The threshold depends on the interface and mesh shape regularity; the stated estimates apply to either subelement.Source: §3 · p. 7
- The error analysis uses hp approximation properties that provide piecewise polynomial approximants in the element spaces.The approximation inequalities have constants depending on smoothness s but independent of h and p.Source: §3 · pp. 7–8
- For s≥2, extension operators map subdomain Sobolev functions into Hs(Ω)∩H1_0(Ω), with the second operator also requiring zero boundary values on Ω2.The extensions apply to functions on Ω1 and to functions on Ω2 that vanish on ∂Ω.Source: §3 · p. 8
4 Error estimates for the symmetric interface penalty finite element method
The symmetric interface penalty method is analyzed through stability and approximation results to obtain broken-H1 and L2 error estimates. The L2 estimate uses Nitsche’s duality argument.
4 Error estimates for the symmetric interface penalty finite element method
- Lemma 4.2 states that a constant α0 exists, independent of h, p, and the penalty parameters, with the stated conditions 0 < γ1 ≲ 1 and γ0 ≥ α0.Source: §4 · p. 10
- For s ≥ 2 and μ = min(p + 1, s), piecewise H^s regularity yields an approximant in V^p.Source: §4 · p. 11
- Under piecewise H^s regularity and the stated penalty conditions, the theorem gives a broken-H1 error estimate.The threshold h0 depends only on the interface and mesh shape regularity.Source: §4 · p. 12
- The L2 error estimate is derived by Nitsche’s duality argument and holds under the broken-H1 theorem’s conditions.Source: §4 · p. 13
5 Error estimates for the non-symmetric interface penalty finite element methods
The section establishes error estimates for the non-symmetric interface penalty method under solution-regularity and penalty-parameter conditions. It also compares its matrix structure and L2 convergence with the symmetric method.
5 Error estimates for the non-symmetric interface penalty finite element methods
- Under piecewise H^s regularity and stated penalty-parameter conditions, the hp-NIPFEM has a broken-H1 error estimate.The theorem assumes s ≥ 2, µ = min {p + 1, s}, 0 < γ1 ≲ 1, and γ0 ≳ 1.Source: §5 · p. 14
- The section derives an L2-error estimate for the hp-NIPFEM, with the theorem presented under the preceding estimate's conditions.The passage identifies the result as an L2-error estimate but does not state its formula.Source: §5 · p. 14
- The NIPFEM stiffness matrix is non-symmetric, and its penalty parameters need only satisfy γ0γ1 ≥ C for any positive C.For the SIPFEM, the corresponding C is problem-dependent.Source: §5 · p. 15
- Unlike the SIPFEM, the NIPFEM's L2-error estimate is not optimal with respect to h.Source: §5 · p. 15
6 Proof of Lemma 3.1
The proof establishes local trace and inverse trace inequalities by constructing regions inside elements intersected by the interface. It handles the two-dimensional geometry in detail and sketches the corresponding three-dimensional construction.
6 Proof of Lemma 3.1
- The lemma's proof establishes local trace and inverse trace inequalities for interface segments or patches within an element.Source: §6 · p. 15
6.1 Two dimensional case
- In 2D, the interface segment is parameterized within a smooth subsegment, and a fan-like region inside the element supports the trace-inequality argument.The proof uses a geometric lower bound on G(ξ) for sufficiently small elements.Source: §6.1 · pp. 16–17
- For the 2D inverse trace inequality, the proof applies a polynomial inverse inequality along rays from the chosen element point.The polynomial degree along the ray is at most p for triangles and 2p for parallelograms.Source: §6.1 · p. 18
6.2 Three dimensional case
- In 3D, the proof sketches the analogous construction using a cone with apex inside the element and curved base on the interface patch.A geometric lower bound on G(ξ, η) is established for sufficiently small elements.Source: §6.2 · pp. 18–19
- The 3D argument parameterizes smooth interface patches and applies the corresponding radial construction to obtain the trace and inverse trace inequalities.Source: §6.2 · pp. 18–20