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Weak Galerkin Methods for Second Order Elliptic Interface Problems
Lin Mu, Junping Wang, Guowei Wei, Xiu Ye, Shan Zhao
TL;DR
Elliptic interface problems with nonsmooth interfaces can produce low-regularity solutions that challenge high-order numerical schemes. The paper develops a weak Galerkin finite element method with flexible interface treatment and reports improved L∞ convergence, including order 1.75 for low-regularity cases. It also proves convergence for higher-order polynomial WG schemes.
Problem
Nonsmooth interfaces produce low-regularity solutions, while rigorous high-order accuracy for such elliptic interface problems remains difficult.
Method
The paper develops a WG-FEM using weak differential operators, discontinuous finite element functions, interface-aligned meshes, and polynomial spaces including Raviart–Thomas elements.
Results
The lowest-order WG-FEM achieves about order 1.75 in the L∞ norm for H2 solutions with C1 or Lipschitz interfaces, while C1 solutions attain nearly second-order convergence.
Takeaways & Limitations
WG-FEM provides accurate numerical approximations for elliptic interface problems with nonsmooth interfaces and low solution regularity, with higher-order schemes theoretically available through higher-degree polynomials.
Takeaways & Limitations
Rigorous convergence analysis for high-order elliptic interface methods on nonsmooth interfaces was not available in the literature, and fourth-order schemes remained limited to special geometries.
Abstract
from arXiv · showhide
Weak Galerkin methods refer to general finite element methods for PDEs in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic partial differential equations (PDEs) with discontinuous coefficients and interfaces. The paper also presents many numerical tests for validating the WG-FEM for solving second order elliptic interface problems. For such interface problems, the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design high order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order one for the solution itself in $L_\infty$ norm. It is demonstrated that the WG-FEM of lowest order is capable of delivering numerical approximations that are of order 1.75 in the usual $L_\infty$ norm for $C^1$ or Lipschitz continuous interfaces associated with a $C^1$ or $H^2$ continuous solutions. Theoretically, it is proved that high order of numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element.
1. Introduction.
The paper addresses elliptic interface problems with discontinuous coefficients, singular sources, and nonsmooth interfaces, where low solution regularity makes high-order schemes difficult. It develops a WG-FEM to flexibly enforce interface conditions and reports numerical improvements over prior results.
- Higher-order convergence is difficult for nonsmooth or Lipschitz interfaces, especially in three-dimensional domains with geometric singularities.
- Low solution regularity near geometric singularities can reduce convergence, motivating methods that solve the original PDE directly rather than regularizing it.
- WG-FEM uses discontinuous functions and weak continuity through discrete differential operators to flexibly handle geometric complexity, boundary conditions, and interface jumps.
- The paper combines a WG formulation, convergence analysis, and numerical tests on smooth, complex, nonsmooth, and low-regularity interfaces.
2. Methods and Algorithms.
The method uses interface-aligned finite element partitions and separate weak Galerkin spaces on the two subdomains. Its discrete functions carry independent interior and boundary polynomial components, with a discrete gradient and Raviart–Thomas spaces supporting the formulation.
- The mesh partition is required to align element edges with the interface and is divided into subdomain-specific element sets.
- Weak Galerkin discrete functions use separate interior and boundary components that may differ on element boundaries.
- The trial and test spaces assign polynomial spaces Pj(K0) and Pℓ(e) to element interiors and edges or faces.
- The discrete gradient maps each weak function into a vector-valued polynomial space Vr(K) on every element.
- The study chooses j = ℓ = m = k and uses the Raviart–Thomas space RTk(K) for Vr(K), while leaving other index combinations for future work.
- The algorithm solves for subdomain weak functions and an interface multiplier subject to projected boundary data and weak variational equations.
3. Convergence Theory.
The convergence theory establishes solvability of the WG scheme and derives error estimates using projection identities and discrete-gradient arguments. The analysis assumes sufficient piecewise regularity and initially takes the coefficient tensor constant on each element.
- Lemma 3.1 proves that the weak Galerkin finite element method has a unique solution.
- Uniqueness is shown by testing the homogeneous system, obtaining zero interface traces, zero discrete gradients, and finally zero weak functions and multiplier.
- The convergence analysis uses local L2 projections, mixed-finite-element projection operators, and an identity relating the discrete gradient to projected fluxes.
- Theorem 3.4 provides error estimates for the WG solution under piecewise H^(k+2) regularity on the two subdomains.
- The proof first assumes that A is constant on each element, while stating that the argument extends to non-constant tensor coefficients.
- The numerical section validates the WG algorithm on benchmark problems using piecewise constant P0(∂K)-P0(K)-RT0(K) elements on structured triangular meshes.
4. Numerical experiments.
Numerical experiments show that the WG method maintains high convergence rates across discontinuous coefficients, high-frequency solutions, nonsmooth interfaces, and low-regularity solutions. The method achieves second-order solution accuracy and first-order gradient accuracy in many cases, with solution rates near 1.75 when function singularities are present.
- Example 1: Piecewise constant WG approximations achieve second-order L∞ convergence for the solution and first-order convergence for its gradient.These rates agree with the theoretical results under uniform triangular mesh refinement.
- Example 1: Mesh refinement substantially improves the piecewise constant WG solution for the circular interface problem.The level-1 solution is visibly piecewise constant, whereas mesh level 5 produces a much better approximation; function and flux jumps remain constant across the interface.
- Example 3: For coefficient contrasts b = 10 and b = 1000, WG retains first-order gradient and second-order solution convergence in the L∞ norm.The larger contrast increases the linear-system condition number and computational effort, yet the method remains robust.
- Nonsmooth interfaces: For C2-continuous solutions across nonsmooth interfaces, WG preserves second-order solution and first-order gradient accuracy without special treatment of interfaces cutting through elements.This behavior is reported for both kinked interfaces and interfaces not aligned with the triangulation.
- Low-regularity interfaces: Across low-regularity tests, WG achieves about 1.75th-order L∞ convergence for H2 solutions while maintaining roughly first-order gradient convergence.For C1 solutions, solution convergence remains close to second order for both C1 and Lipschitz interfaces; function singularities, rather than geometric singularities, cause the main reduction.
5. Conclusion.
For Lipschitz continuous interfaces with C^1 or H^2 continuous solutions, the lowest-order WG-FEM achieves higher convergence rates than previously reported results. The paper also establishes that higher-order schemes can be designed using higher-degree element polynomials.
- 1.75-order convergence in the solution is achieved by the lowest-order WG-FEM for Lipschitz continuous interfaces with C^1 or H^2 continuous solutions.
- 1-order convergence in the gradient is achieved under the same interface and solution regularity conditions.
- Higher-order WG methods can be designed using high-order polynomials on each element.