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A Weak Galerkin Finite Element Method with Polynomial Reduction
Lin Mu, Junping Wang, Xiu Ye
TL;DR
The paper addresses how to reduce polynomial unknowns in weak Galerkin methods without sacrificing convergence. It proposes a lower-degree boundary-space configuration for second-order elliptic problems and analyzes its discrete weak-gradient scheme. The method achieves optimal-order convergence theoretically and numerically across polygonal and polyhedral meshes.
Problem
The paper asks whether polynomial spaces can be combined to reduce unknowns without compromising the convergence order of WG methods.
Method
The paper proposes a WG scheme using (P_k(T), P_{k−1}(e), P_{k−1}(T)^d) and discrete weak gradients for second-order elliptic problems.
Results
Optimal-order convergence is established in H^1 and L^2 norms, with rates O(h^k) and O(h^{k+1}), and confirmed numerically for k = 1, 2 across several mesh types.
Takeaways & Limitations
The reduced WG formulation retains the reported convergence order while supporting flexible polygonal and polyhedral finite element partitions.
Takeaways & Limitations
The error analysis assumes a piecewise constant coefficient tensor, although extension to piecewise sufficiently smooth variable tensors is stated to be possible.
Abstract
from arXiv · showhide
The novel idea of weak Galerkin (WG) finite element methods is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different weak Galerkin finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. For illustrative purpose, the authors use second order elliptic problems to demonstrate the basic idea of polynomial reduction. A new weak Galerkin finite element method is proposed and analyzed. This new finite element scheme features piecewise polynomials of degree $k\ge 1$ on each element plus piecewise polynomials of degree $k-1\ge 0$ on the edge or face of each element. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^1$ norm and the standard $L^2$ norm. In addition, the paper presents a great deal of numerical experiments to demonstrate the power of the WG method in dealing with finite element partitions consisting of arbitrary polygons in two dimensional spaces or polyhedra in three dimensional spaces. The numerical examples include various finite element partitions such as triangular mesh, quadrilateral mesh, honey comb mesh in 2d and mesh with deformed cubes in 3d. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.
1. Introduction.
The paper seeks polynomial spaces that reduce unknowns without sacrificing convergence, introducing a WG scheme with lower-degree edge or face polynomials for second-order elliptic problems.
- The paper explores polynomial-space combinations that minimize unknowns without compromising convergence order.
- Earlier WG formulations use discrete weak gradients with RT or BDM spaces but are limited to triangular or tetrahedral partitions.
- The proposed configuration is (P_k(T), P_{k-1}(e), P_{k-1}(T)^d) for the element, edge or face, and gradient spaces.
- The corresponding WG solution converges at O(h^k) in H^1 and O(h^{k+1}) in L^2.
- Numerical tests report optimal-order convergence for k = 1, 2 on triangular, quadrilateral, honeycomb, and deformed-cube meshes.
- The paper develops weak-gradient definitions, the new scheme, error estimates, and numerical tests for polygonal or polyhedral meshes.
2. Weak Gradient and Discrete Weak Gradient.
WG represents functions by interior and boundary components and defines weak gradients distributionally; the discrete weak gradient is their polynomial approximation determined by an integration-by-parts identity.
- A weak function on an element is represented as v = {v_0, v_b}, with separate interior and boundary components.
- The boundary component v_b need not equal the trace of the interior component v_0 when a trace is defined.
- The weak gradient is a linear functional on H(div,K), defined through interior and boundary pairings with test fields.
- For smooth H^1 functions, the weak gradient coincides with the strong gradient.
- The discrete weak gradient is the unique polynomial in [P_r(K)]^d satisfying the corresponding weak-gradient identity for every polynomial test field.
- Integration by parts rewrites the discrete identity in terms of the interior gradient and the boundary mismatch v_b − v_0.
3. Weak Galerkin Finite Element Schemes.
The new WG scheme uses degree-k interior polynomials and degree-(k−1) boundary polynomials, computes a discrete weak gradient, and solves a stabilized symmetric positive definite system.
- The WG space uses v_0 of degree k on each element and v_b of degree k−1 on each edge or face.
- Each edge or face has a single shared boundary value v_b across adjacent elements.
- The discrete weak gradient is computed locally in [P_{k−1}(T)]^d using the weak-gradient definition.
- The method combines a weak-gradient bilinear form with a stabilization penalizing projected interior-boundary discrepancies.
- The numerical approximation satisfies the stabilized variational equation with prescribed projected boundary data.
- The resulting system is symmetric and positive definite for every stabilization parameter ρ > 0 and has a unique solution.
4. Mass Conservation.
The WG scheme preserves mass through an appropriately defined numerical flux whose normal component is continuous across interior edges or faces.
- The mass-conservation argument uses an integral form obtained by integrating the governing equation over each control volume.
- The WG approximation retains mass conservation with an appropriately defined numerical flux.The property is established by testing the discrete scheme with elementwise constant functions.
- The numerical flux has a continuous normal component across interior edges or faces.Opposite outward normals on neighboring elements yield continuity of qh · n.
5. Some Technical Estimates.
The paper establishes projection, norm-equivalence, and approximation estimates on shape-regular polygonal or polyhedral meshes to support its error analysis.
- Role in analysis: The technical lemmas provide bounds for projection operators and weak functions needed in the forthcoming error analysis.The paper explicitly introduces these results as prerequisites for error estimates.
- Projection estimates: The discrete weak gradient commutes with the L2 projections: ∇w(Qhφ) = Qh(∇φ).This identity follows from the weak-gradient definition, integration by parts, and the projection definitions.
- Mesh assumptions: The estimates apply to sufficiently general, shape-regular partitions that allow polygons or polyhedra.The analysis uses trace and inverse inequalities and L2 projection approximation properties.
- Norm estimates: The discrete H1 semi-norm is equivalent to the trip-bar norm on the finite element space.There are positive constants C1 and C2 such that C1∥v∥1,h ≤ |||v||| ≤ C2∥v∥1,h.
6. Error Analysis.
The error analysis derives optimal-order WG estimates in discrete H1 and standard L2 norms under solution regularity and, for L2, dual-problem regularity assumptions.
- Assumptions: The analysis assumes the coefficient tensor a is piecewise constant, with extension to piecewise sufficiently smooth variable tensors.
- Error estimates: Under u ∈Hk+1(Ω), the projected-solution error satisfies |||uh −Qhu||| ≤ Ch^k∥u∥k+1.
- Error estimates: Under u ∈Hk+1(Ω), the discrete H1 error satisfies ∥u −uh∥1,h ≤ Ch^k∥u∥k+1.
- L2 analysis: The standard L2 error estimate is derived with a duality argument assuming the dual problem has usual H2-regularity.
- Numerical results: Numerical tests use k = 1 and k = 2 elements on triangular, deformed rectangular, honeycomb, and deformed cubic meshes.These experiments examine convergence and flexibility in two and three dimensions.
7. Numerical Examples.
Numerical experiments test WG convergence across polynomial orders and mesh geometries, including triangular, quadrilateral, honeycomb, and deformed cubic meshes. The results report optimal or theoretically predicted convergence rates in H1 and L2 norms.
- Experimental setup: The experiments measure WG errors using discrete H1 and L2-related norms for elliptic problems.The error is defined from the WG solution and L2 projections onto the relevant spaces.
- Triangular meshes: Linear WG elements on triangular meshes converge at O(h) in H1 and O(h2) in L2 norms.This rate is reported for the lowest-order case k = 1.
- Triangular meshes: For triangular meshes, k = 1 and k = 2 configurations exhibit optimal numerical convergence rates in H1 and L2 norms.The configurations are denoted (P1(T), P0(e)) and (P2(T), P1(e)).
- Quadrilateral meshes: Quadrilateral-mesh experiments with k = 1 and k = 2 also show optimal convergence rates in H1 and L2 norms.The meshes are successively refined by connecting element barycenters with edge midpoints.
- Honeycomb and deformed cubic meshes: Honeycomb and deformed cubic mesh tests confirm convergence behavior predicted by the theory.The deformed cubic results achieve O(h) in H1 and O(h2) in L2 norms.