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Partially Penalized Immersed Finite Element Methods for Elliptic Interface Problems
Tao Lin, Yanping Lin, Xu Zhang
TL;DR
Classic immersed finite element methods can lose accuracy near discontinuous interface edges and may deteriorate on finer meshes. This paper introduces partially penalized IFE methods with interface-edge stabilization, proves optimal H1 convergence under sufficient regularity, and reports stable H1 and L2 convergence as meshes are refined.
Problem
Trace-inequality constants may depend on interface location, while classic IFE methods can exhibit large interface-element errors and deteriorating convergence on finer meshes.
Method
The paper develops partially penalized IFE methods on Cartesian meshes, adding stabilization terms only on interface edges to control discontinuities in IFE functions.
Results
The methods are proven to achieve optimal H1 convergence under sufficient exact-solution regularity, while numerical results show maintained H1 and L2 convergence on finer meshes.
Takeaways & Limitations
Trace inequalities for linear and bilinear IFE functions support the error analysis, and the methods retain the classic IFE methods’ global degrees of freedom.
Takeaways & Limitations
Classic IFE methods remain known to have poor point-wise accuracy around interfaces, motivating but not resolving the broader accuracy challenge across all settings.
Abstract
from arXiv · showhide
This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE functions. With the enhanced stability due to the added penalty, not only these IFE methods can be proven to have the optimal convergence rate in the H1-norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the H1-norm and the L2-norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic IFE methods in some situations. Trace inequalities are established for both linear and bilinear IFE functions that are not only critical for the error analysis of these new IFE methods, but also are of a great potential to be useful in error analysis for other IFE methods.
1. Introduction.
The article develops partially penalized IFE methods for elliptic interface problems on Cartesian meshes, targeting discontinuities across interface edges and associated error-estimation difficulties. The methods preserve classic IFE degrees of freedom while supporting optimal convergence and extensions toward DG formulations.
- The proposed methods solve elliptic interface problems on Cartesian meshes, including interfaces with nontrivial geometries.
- Error estimation is difficult on edges between interface elements because IFE functions can be discontinuous there.
- Penalty terms are introduced on interface edges to control discontinuity-related effects and address convergence deterioration observed for classic IFE methods.
- The partially penalized methods retain the same global degrees of freedom as their classic counterparts, whereas DG IFE methods may have about 6 times more unknowns on Cartesian triangular meshes.
- The methods and analysis can be modified for DG IFE formulations, which offer adaptivity even on Cartesian meshes.
- The article establishes trace inequalities, proves optimal energy-norm convergence, and presents numerical examples for the proposed schemes.
2. Partially penalized IFE methods.
The methods construct linear or bilinear IFE spaces on Cartesian meshes and add interface-edge stabilization to their weak formulation. The resulting solution remains continuous at mesh vertices while permitting the interface-related discontinuities addressed by the method.
- Interior edges are classified as interface or non-interface edges, with averages and jumps defined for functions across shared edges.
- The analysis assumes sufficient regularity so the exact solution belongs to the piecewise Sobolev space ˜H2(Ω).
- The weak formulation is obtained by multiplying the governing equation by a test function, integrating elementwise, and applying Green’s formula.
- The construction uses linear or bilinear IFE basis functions on interface elements and standard finite-element basis functions on non-interface elements.
- The IFE space requires elementwise membership, continuity at mesh vertices, and homogeneous boundary values.
- The partially penalized IFE method seeks uh in the global IFE space using the weak form and retains the stated finite-element continuity constraints.
3. Trace inequalities for IFE functions.
The section extends trace inequalities to linear and bilinear IFE functions on interface elements, establishing constants independent of interface location. These results address the difficulty caused by discontinuities and the lack of H2 regularity on interface elements.
- Standard trace inequalities extend to linear IFE functions because their local space is contained in C(K) ∩ H1(K).
- The second standard trace inequality cannot directly apply to interface-element IFE functions because they generally do not belong to H2(K).
- 3.1. Trace inequalities for linear IFE functions.: The same interface-location-independent trace bounds are established for linear IFE functions on rectangular interface elements.
- 3.1. Trace inequalities for linear IFE functions.: A constant independent of interface location bounds weighted function traces and normal fluxes for linear IFE functions on triangular interface elements.
- 3.2. Trace inequalities for bilinear IFE functions.: For bilinear IFE functions, trace-inequality proofs are more complicated because their gradients are not constant.
- 3.2. Trace inequalities for bilinear IFE functions.: A constant independent of interface location also bounds weighted function traces and normal fluxes for bilinear IFE functions on rectangular interface elements.
4. Error Estimation for Partially Penalized IFE Methods.
The analysis establishes coercivity for partially penalized IFE bilinear forms and combines trace inequalities with interpolation estimates to derive error bounds for linear and bilinear IFE solutions.
- Coercivity: Coercivity follows directly for ϵ = 1 and requires sufficiently large stabilization parameters for ϵ = 0 or ϵ = −1.For ϵ = 0, σ0B = C; for ϵ = −1, σ0B = 5C/2.
- Trace inequalities: Trace inequalities for linear and bilinear IFE functions control interface-edge terms in the error analysis.These inequalities are applied to bound terms involving the two elements sharing an interface edge.
- Interpolation estimates: Under sufficient regularity, interpolation in linear or bilinear IFE spaces provides the approximation estimates needed for solution-error bounds.The analysis uses optimal approximation capability and an additional interface-edge interpolation estimate for u ∈ ˜H3(Ω).
- Error bounds: For α = 1 and u ∈ ˜W 2,∞(Ω), the analysis also establishes an error estimate using the interface-edge interpolation bound.The corresponding result is stated as Remark 4.2.
5. Numerical Examples.
The numerical examples compare partially penalized and classic IFE methods on Cartesian meshes for elliptic interface problems, including large and moderate coefficient jumps. Partially penalized methods retain optimal H1 convergence on refined meshes and show improved L2, L∞, and point-wise behavior in the moderate-jump case.
- Experimental setup: The experiments use a rectangular domain with a circular interface and Cartesian meshes formed from N × N congruent squares.The interface has radius r0 = π/6.28, and the mesh size is h = 2/N.
- Large coefficient jump: For a large coefficient jump (β−, β+) = (1, 10000), all partially penalized methods converge optimally in the H1-norm.The classic IFE method also converges optimally in the H1-norm in this example.
- Moderate coefficient jump: For a moderate jump (β−, β+) = (1, 10), partially penalized methods maintain the predicted O(h) H1 convergence rate across all tested meshes.The classic IFE method slightly loses its H1 convergence rate on the finest meshes.
- Moderate coefficient jump: In the moderate-jump case, partially penalized methods converge optimally in L2, while the classic IFE method’s L2 rate clearly degenerates as the mesh becomes finer.A similar L∞ convergence phenomenon is reported for the moderate coefficient jump.
- Point-wise errors: On an 80 × 80 rectangular mesh, the classic bilinear IFE solution has much larger errors near the interface than the nonsymmetric partially penalized solution.The nonsymmetric partially penalized solution has uniformly smaller error magnitude over the solution domain, and similar behavior is observed for other partially penalized solutions.
- Supporting analysis: The numerical section also includes trace-inequality proofs for linear IFE functions, with the same arguments stated to apply to bilinear IFEs.The proof treats an interface triangle and estimates an interface edge; the other interface edge is handled similarly.