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Weak Galerkin Finite Element Methods on Polytopal Meshes
Lin Mu, Junping Wang, Xiu Ye
TL;DR
Second-order elliptic finite element methods commonly use restricted mesh shapes, while nonconforming spaces make classical gradients problematic. This paper uses a stabilized weak-gradient method on polytopal meshes, establishes an optimal-order discrete-norm error estimate, and reports numerical results confirming its theory.
Problem
Method
The method applies a discrete weak gradient and stabilization to polynomial weak-function spaces on shape-regular polytopal partitions.
Source: §1 · p. 3, §3 · pp. 4–5Results
The discrete H1-equivalent norm error is O(h^k) for solutions in H^(k+1)(Ω), and the paper reports numerical results confirming its theory.
Source: §1 · p. 4, §8.2 · p. 14Takeaways & Limitations
The scheme supports arbitrary shape-regular polytopal meshes, computation-friendly polynomial spaces, and local mass conservation.
Source: §1 · p. 3For the degenerate test problem with coefficient a = xy, the earlier convergence theory does not apply without modification.
Source: §9.2 · p. 18
Abstract
from arXiv · showhide
This paper introduces a new weak Galerkin (WG) finite element method for second order elliptic equations on polytopal meshes. This method, called WG-FEM, is designed by using a discrete weak gradient operator applied to discontinuous piecewise polynomials on finite element partitions of arbitrary polytopes with certain shape regularity. The paper explains how the numerical schemes are designed and why they provide reliable numerical approximations for the underlying partial differential equations. In particular, optimal order error estimates are established for the corresponding WG-FEM approximations in both a discrete $H^1$ norm and the standard $L^2$ norm. Numerical results are presented to demonstrate the robustness, reliability, and accuracy of the WG-FEM. All the results are derived for finite element partitions with polytopes. Allowing the use of discontinuous approximating functions on arbitrary polytopal elements is a highly demanded feature for numerical algorithms in scientific computing.
1. Introduction.
The paper develops a weak Galerkin method for second-order elliptic problems on general polytopal meshes. Its design combines weak gradients with stabilization to support flexible meshes while retaining local mass conservation.
1. Introduction.
- Weak Galerkin methods replace strong derivatives in variational forms with weakly defined differential operators, here approximated through a local weak gradient.This addresses the difficulty that classical gradients are not well-defined for non-H1 trial functions.Source: §1 · p. 2
- Earlier WG methods based on RT or BDM elements were limited to triangular or tetrahedral partitions, while a mixed WG method already admitted shape-regular polytopes [17] [18].Source: §1 · pp. 2–3
- The method applies stabilization to a weak-gradient variational form to handle arbitrary polytopal finite element partitions in the primary variable.The authors note that this flexibility is useful for complex geometries and adaptive mesh refinement.Source: §1 · pp. 2–3
- Its three stated contributions are shape-regular polytopal meshes, computation-friendly regular polynomial spaces on elements and faces, and local mass conservation.Source: §1 · p. 3
- For the Poisson equation, this WG method is identical to HDG, but their concepts differ: WG centers on weak gradients and HDG on fluxes.The authors state that WG avoids inverting the coefficient matrix for nonlinear or degenerate coefficients.Source: §1 · p. 3
- The paper reports optimal-order discrete H1- and L2-norm error estimates and numerical results that confirm the developed theory.Source: §1 · pp. 3–4
2. Preliminaries and Notations.
This section establishes Sobolev-space notation used in the paper. It identifies H0 with L2 and specifies the vector-field space H(div; D).
2. Preliminaries and Notations.
- For integer s, the section uses the standard Sobolev spaces, inner products, norms, and seminorms, with multi-index notation for derivatives.Source: §2 · p. 4
- H0(D) is identified with L2(D), and when D is the domain Ω, the domain subscript is omitted from norms and inner products.Source: §2 · p. 4
- H(div; D) consists of vector-valued functions that, together with their divergence, are square integrable.Source: §2 · p. 4
3. Weak Gradient.
The section defines a discrete weak gradient to replace strong derivatives in the variational formulation. It applies this operator to weak functions whose interior and boundary values are independent components.
3. Weak Gradient.
- Weak Galerkin methods use discrete weak derivatives in place of strong derivatives in variational forms, and this section develops the gradient operator for the model problem.Source: §3 · p. 4
- A weak function on an element K has an interior component v0 and a boundary component vb, which need not be related by a trace.Source: §3 · p. 4
- The weak gradient acts on H(div, K) through the interior divergence term and the boundary normal-flux term.For smooth H1 functions, the weak gradient agrees with the strong gradient.Source: §3 · p. 5
- The discrete weak gradient is the polynomial in [Pr(K)]^d satisfying the weak-gradient identity for every polynomial test vector in that space.This broader choice of polynomial subspace provides more flexibility than the RT- or BDM-based examples used in earlier WG work [17].Source: §3 · p. 5
4. Weak Galerkin Finite Element Schemes.
The method is designed for shape-regular polygonal and polyhedral meshes, with discrete weak gradients and discontinuous polynomial functions. Its algorithm combines these ingredients with stabilization to approximate the elliptic problem.
4. Weak Galerkin Finite Element Schemes.
- The WG scheme uses discontinuous piecewise polynomials and a discrete weak gradient on general polygonal and polyhedral meshes.The method is designed to allow arbitrary polytopal elements rather than being restricted to RT or BDM elements for computing the discrete weak gradient.Source: §4 · p. 5, §4.2 · p. 7
4.1. Domain Partition.
- The mesh consists of closed, simply connected polygons or polyhedra satisfying shape-regularity conditions A1–A4.These conditions include flat edges or faces and geometric requirements on pyramids above each face and on circumscribed simplices.Source: §4.1 · p. 6
4.2. WG Finite Element Algorithms.
- For each element, the WG algorithm uses degree-k interior and boundary polynomials and solves a stabilized variational equation with prescribed boundary data.The stabilization parameter ρ is positive and may be set to 1 in practical computation.Source: §4.2 · pp. 7–8
5. L2 Projection Operators.
This section defines projections onto the discrete weak-function and gradient spaces and establishes their approximation properties. The projection commutes with the discrete weak gradient on each element.
5. L2 Projection Operators.
- The projection Q_h commutes with the discrete weak gradient: ∇w(Q_hφ) = Q_h(∇φ) on each element.Q_h projects the interior and boundary components separately, while the gradient projection is onto the local discrete gradient space.Source: §5 · p. 8
- For shape-regular polygonal or polyhedral meshes, the section establishes approximation estimates for Q_h when φ ∈ H^(k+1)(Ω).The projection lemma’s proof relies on technical tools established for general meshes in [18].Source: §5 · p. 8
- Lemma 5.3 gives a result assuming a shape-regular mesh, for any w ∈ H^(k+1)(Ω) and v = {v0, vb} ∈ V_h.Source: §5 · p. 9
6. On Mass Conservation.
The WG approximation retains mass conservation when expressed using an appropriately defined numerical flux. Its normal component is continuous across element boundaries.
6. On Mass Conservation.
- The WG approximation retains the mass conservation property with an appropriately defined numerical flux.The proof tests the discrete equation with an elementwise constant interior component and zero boundary component; the resulting flux is given in the section.Source: §6 · pp. 9–10
- The numerical flux’s normal component is continuous across the boundary of each element.On a shared interior face, the two outward normals sum to zero, yielding the stated continuity.Source: §6 · p. 10
7. Existence and Boundedness of WG Solutions.
The section establishes existence and boundedness for linearized WG problems, and existence for the general nonlinear scheme. Its analysis uses a triple-bar norm that is equivalent to H1 for finite element functions with vanishing boundary values.
7. Existence and Boundedness of WG Solutions.
- The coefficient assumptions make the norms associated with different linearization states uniformly equivalent.Source: §7 · p. 11
- For finite element functions with vanishing boundary values, the triple-bar norm is H1-equivalent and satisfies a Poincaré-type inequality on shape-regular partitions.Source: §7 · p. 11
- The linearized scheme has a unique solution with a bound controlled by the forcing and the least-norm admissible boundary lift.The bound is |||uh|||φ ≤ C(∥f∥ + infψ∈Bg,h |||ψ|||).Source: §7 · p. 12
- The general nonlinear WG scheme has at least one solution, established using the Leray–Schauder fixed point theorem.The theorem’s boundedness condition is verified for the associated mapping.Source: §7 · p. 13
8. Error Analysis.
For the linear elliptic case, the analysis derives an error equation and proves optimal-order error estimates under solution regularity assumptions. The L2 estimate additionally assumes H2 regularity for the dual problem.
8. Error Analysis.
- The convergence analysis is restricted to linear coefficients; piecewise-constant tensors are assumed, with extension to variable tensors requiring piecewise sufficient smoothness.Source: §8 · p. 14
8.1. Error equation.
- Subtracting the WG scheme from the tested continuous-problem identity yields an error equation for the difference between the WG solution and the L2 projection of the exact solution.Source: §8.1 · p. 14
8.2. Error estimates.
- |||uh − Qhu||| ≤ C h^k ∥u∥k+1 for u ∈ Hk+1(Ω), giving the stated optimal-order estimate in the discrete H1 norm.Source: §8.2 · pp. 14, 16
- The standard L2 error estimate is obtained under exact-solution regularity and the usual H2 regularity of the dual problem.The analysis introduces a dual problem to derive this estimate.Source: §8.2 · p. 15
9. Numerical Experiments.
The experiments examine convergence rates and accuracy across uniform, degenerate, deformed, and hanging-node mesh settings. They use piecewise linear functions and assess WG solutions in multiple norms.
9. Numerical Experiments.
- The experiments use piecewise linear functions and measure WG-solution errors in three norms, including element-based and edge-based L2 norms.The section examines convergence rates and accuracy on polyhedral meshes with and without hanging nodes.Source: §9 · p. 17
9.1. Case 1: Poisson Problem on Uniform Meshes.
- On uniform rectangular and triangular meshes, the k = 1 WG solution converges at O(h) in H1 and O(h2) in L2.The Poisson test uses a square domain and a known exact solution.Source: §9.1 · pp. 17–18
9.2. Case 2: Degenerate Elliptic Problems.
- For the degenerate elliptic problem, the new WG-FEM has optimal-order convergence in H1 and L2, while the two comparison methods converge at significantly lower orders.The coefficient a = xy vanishes at the origin; the comparison includes interior penalty DG P1 and an earlier WG method with piecewise constants.Source: §9.2 · pp. 18–19, Table 9.3 · p. 18
9.3. Case 3: WG-FEM on Deformed Rectangular Meshes.
- On successively refined deformed rectangular meshes, WG-FEM exhibits optimal-order convergence in various norms.The initial mesh is deformed, and refinement connects each element barycenter to the midpoints of its edges.Source: §9.3 · p. 19, Figure 9.1 · p. 19, Table 9.4 · p. 19
9.4. Case 4: WG-FEM on Meshes with Hanging Nodes.
- For hanging-node meshes, the method partitions affected edges at hanging nodes and uses piecewise linear edge functions; this refinement may produce elements outside the stated shape-regularity condition.The section reports convergence results for the resulting mesh setting in Table 9.5.Source: §9.4 · pp. 19–20, Table 9.5 · p. 20, Figure 9.2 · p. 20