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Weak Galerkin Finite Element Methods for the Biharmonic Equation on Polytopal Meshes
Lin Mu, Junping Wang, Xiu Ye
TL;DR
The paper addresses the difficulty of constructing practical finite elements for biharmonic equations on general meshes. It develops a weak Galerkin method using generalized or discontinuous polynomial functions and analyzes its convergence. The method achieves optimal-order convergence in a discrete H2 norm, while L2 convergence is sub-optimal for quadratic elements and optimal for cubic or higher-order elements.
Problem
Constructing C1-continuous piecewise-polynomial elements for H2-conforming biharmonic methods is difficult, motivating methods that work on general polygonal or polyhedral partitions.
Method
The paper develops a weak Galerkin finite element method based on generalized or discontinuous approximating functions, a weak Laplacian, and a stabilizer on shape-regular polygonal or polyhedral meshes.
Results
The analysis establishes optimal-order convergence in a discrete H2 norm; in L2, quadratic elements achieve O(h2), while cubic or higher-order elements achieve O(hk+1).
Takeaways & Limitations
Weak Galerkin discretizations provide a flexible biharmonic finite element framework for general polygonal or polyhedral meshes while retaining the stated convergence guarantees.
Takeaways & Limitations
The paper calls for more numerical experiments, especially for high-order WG elements, and identifies hybridization to eliminate element-local interior unknowns as future work.
Abstract
from arXiv · showhide
A new weak Galerkin (WG) finite element method is introduced and analyzed in this paper for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding WG finite element solutions. Error estimates in the usual $L^2$ norm are also derived, yielding a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions.
1. Introduction.
The paper develops a weak Galerkin method for biharmonic equations that avoids the construction of difficult C1-continuous elements while supporting general polygonal and polyhedral meshes. It specifies the method and establishes a convergence theory.
- Motivation: H2-conforming methods require C1-continuous piecewise polynomials, making practical construction difficult for biharmonic problems.The paper notes that such methods are rarely used in practice because of this construction complexity.
- Related approaches: Existing alternatives include nonconforming, discontinuous Galerkin, interior-penalty, and mixed finite element methods for biharmonic equations.The related approaches include the Morley element, C0 interior penalty, hp-version interior-penalty discontinuous Galerkin, and mixed formulations.
- Proposed approach: The proposed weak Galerkin method uses generalized or discontinuous approximating functions on polygonal or polyhedral partitions of arbitrary shape with suitable shape regularity.This design is intended to overcome the barrier posed by constructing smooth finite element functions.
- Proposed approach: The method replaces the Laplacian with a discrete weak Laplacian and uses a stabilizer to enforce weak continuity of approximating functions.A direct replacement of the differential operator is described as insufficient without this stabilizing mechanism.
- Contributions: The paper specifies the WG formulation and justifies it by establishing a mathematical convergence theory.The stated goal includes detailing the formulation and proving its rigor through convergence analysis.
2. Preliminaries and Notations.
This section introduces Sobolev-space notation, multi-index derivatives, inner products and norms, and the space H(div; D) used later in the analysis.
- Sobolev notation: For an open bounded Lipschitz domain D, the paper uses standard Sobolev spaces Hs(D) with associated inner products, norms, and seminorms.The notation applies for s ≥ 0 and includes the usual integer-order seminorm definition.
- Sobolev notation: A multi-index α is represented by its components, total order |α|, and derivative operator ∂α.These quantities support the notation for integer-order Sobolev seminorms.
- Notation conventions: When D is the domain Ω, subscripts are omitted from norm and inner-product notation.The paper identifies H0(D) with L2(D) and denotes its norm and inner product by ∥·∥D and (·,·)D.
- Function spaces: H(div; D) consists of vector-valued functions whose components and divergence are square integrable.Its norm combines the L2 norm of the vector field with the L2 norm of its divergence.
3. Weak Laplacian and Discrete Weak Laplacian.
The paper defines weak and discrete weak Laplacians for generalized functions with interior, boundary-value, and boundary-gradient components. The discrete operator is a polynomial approximation of the weak Laplacian.
- Motivation: The biharmonic method targets the Laplacian as its principal differential operator and introduces both weak and discrete versions for numerical computation.The discrete version is defined in polynomial subspaces for practical approximation.
- Weak functions: A weak function on an element T has interior value v0, boundary value vb, and boundary-gradient component vg.The boundary components need not equal the traces of v0 and ∇v0, respectively.
- Weak Laplacian: The weak Laplacian is defined as a linear functional on G2(T) through interior and boundary pairings involving v0, vb, and vg.Its action combines (v0, Δϕ)T with boundary terms involving the outward normal n.
- Weak Laplacian: For functions in H2(T), the weak Laplacian coincides with the strong Laplacian.The paper states Δwv = Δv for all functions identified with H2(T) functions.
- Discrete weak Laplacian: The discrete weak Laplacian Δw,r,T v is the unique polynomial in Pr(T) that approximates the weak Laplacian through the same variational action.The test space consists of polynomials of degree no more than r.
4. Weak Galerkin Finite Element Schemes.
The paper formulates weak Galerkin schemes for the biharmonic equation on shape-regular polygonal or polyhedral meshes, using discrete weak Laplacians, stabilizers, and generalized finite element functions. The schemes define a positive norm, admit unique solutions, and include a reduced-unknown variant.
- Finite element spaces: Shape-regular partitions consist of polygons in 2D or polyhedra in 3D, with weak finite element functions assembled across element interfaces.The local space uses interior polynomials, boundary traces, and gradient-related edge or face components.
- Algorithm I: The quantity |||v|||2 = (∆wv, ∆wv)h + s(v, v) defines a norm on Vh^0, supporting positivity of the formulation.The positivity argument uses vanishing discrete weak Laplacian and stabilizer terms to establish the required kernel properties.
- Algorithm I: The first weak Galerkin scheme has a unique solution.Uniqueness follows by applying the homogeneous problem to the difference of two solutions and using the norm result.
- Algorithm II: A second scheme reduces the number of unknowns by retaining only the normal component of vg, while preserving a unique solution in its corresponding space.Its local functions use v0 in Pk(T), vb in Pk(e), and a scalar normal-gradient component of degree k−1.
5. L2 Projections and Approximation Properties.
The analysis constructs local L2 projections and establishes their compatibility with the classical and discrete weak Laplacians. These projection and trace estimates provide the approximation tools used in the WG convergence analysis.
- Projection operators: The local projections Q0, Qb, and Qg map onto Pk(T), Pk(e), and [Pk−1(e)]d, respectively, for k ≥ 2.The projection of a smooth function into Vh combines the element, boundary, and gradient-trace projections.
- Commutative property: The L2 projections satisfy a commutative property linking the discrete weak Laplacian of a projected function to the projection of its classical Laplacian.Specifically, the discrete weak Laplacian of Qhu equals Qh(∆u) under the stated local assumptions.
- Role in analysis: The derived projection and trace estimates are used in the convergence analysis of both WG finite element schemes.The error analysis presented later focuses on the first scheme and can be modified for the second.
6. An Error Estimate in H2.
Under sufficient regularity, the WG approximation achieves optimal-order convergence in a discrete H2-equivalent norm. The estimate follows from the error equation, the commutative projection property, and approximation bounds.
- Proof strategy: The consistency calculation uses integration by parts, the identity ∆wQhu = Qh(∆u), and the biharmonic equation to connect the discrete formulation with the continuous problem.Boundary trace conditions remove the relevant boundary contributions in the summed element identities.
- Proof strategy: The error equation is obtained by subtracting the WG scheme from the projected continuous equation and using the discrete weak Laplacian identity.The proof then tests with the error and applies Cauchy–Schwarz together with projection estimates.
- Error estimate: The WG solution therefore has optimal-order convergence in the discrete H2-equivalent norm.The paper identifies the norm as essentially an H2 norm on Vh^0.
7. An Error Estimate in L2.
The paper derives an L2 error estimate using a dual problem, regularity assumptions, and the previously established discrete H2 estimate. Quadratic elements converge sub-optimally, while cubic and higher-order elements achieve optimal order.
- Duality argument: The L2 analysis introduces a dual problem and assumes its H4 regularity to control the error.The proof tests the dual problem with the error and combines resulting bounds with the discrete H2 estimate.
- Theoretical estimate: Theorem 7.1 gives the estimate ∥Q0u − u0∥ ≤ C h^(k+t0−2)(∥u∥_(k+1) + h^δ_(k,2)∥u∥_4), with t0 = min(k, 3).The result applies for k ≥ 2 when u ∈ H^(k+1)(Ω) and the dual problem has H4 regularity.
- Convergence orders: Quadratic elements have sub-optimal L2 convergence O(h2), whereas cubic and higher-order elements have optimal convergence O(h^(k+1)).This distinction follows directly from the value of t0 in the theorem.
- Error analysis: The error analysis bounds six terms using Cauchy–Schwarz inequalities and approximation estimates before combining them into the final estimate.The bounds include the stabilizer term and estimates involving t0 = min(k, 3).
- Conclusion: The final L2 estimate is obtained by combining the duality bounds with the previously established H2 error estimate.The proof explicitly identifies the H2 estimate as the final ingredient.
8. Numerical Results.
The numerical section implements a quadratic WG finite element space and locally computes the discrete weak Laplacian. Experiments on triangular meshes report the predicted H2 and L2 rates and confirm the earlier convergence theory.
- Discretization: The numerical scheme uses v0 ∈ P2(T), vb ∈ P2(e), and vg ∈ P1(e) on each element and edge.The method measures errors using an element-based L2 norm and a discrete H2-related norm.
- Discrete weak Laplacian: The discrete weak Laplacian is computed locally in P0(T) from the interior, boundary, and normal-gradient components.Because ψ ∈ P0(T), the defining relation simplifies to a boundary integral involving vg · n.
- Test Case 1: For the homogeneous-boundary test on uniform triangular meshes, the H2 and L2 convergence rates are O(h) and O(h2), respectively.The exact solution is u = x2(1 − x)2 y2(1 − y)2 on the unit square, with h = 1/n.
- Test Case 2: The nonhomogeneous-boundary test uses u = sin(πx) sin(πy), and its Table 8.2 results confirm the earlier convergence theory.The test uses triangular mesh triangulations constructed as in the first case.
- Future work: The authors identify further experiments for both WG schemes and high-order elements as needed, alongside local elimination of v0 through hybridization.These are presented as directions for extending the numerical study and implementation.
A.1. Domain Partition and Shape Regularity.
The appendix defines shape-regular polygonal or polyhedral partitions through geometric conditions on elements, faces, circumscribed simplices, and interior pyramids.
- Mesh domain: The domain partition consists of closed, simply connected polygons in two dimensions or polyhedra in three dimensions.The mesh uses Eh for edges or flat faces, with separate notation for interior faces and element diameters.
- Illustration: Figure A.1 depicts the shape-regular polygonal element used to illustrate these geometric assumptions.The depicted element is labeled ABCDEF A.
- Shape regularity: Shape regularity requires geometric bounds involving element and face sizes, with constants independent of individual elements.The assumptions include conditions labeled A1 and A2.
- Geometric conditions: Mesh edges and faces must be flat, and every face must support an interior pyramid whose height is proportional to the element diameter.The proportionality factor satisfies σe ≥ σ∗ > 0, together with an angle condition.
- Circumscribed simplices: Each element must have a shape-regular circumscribed simplex with diameter comparable to the element diameter.The circumscribed simplices may intersect only a fixed and small number of other such simplices.
A.2. A Trace Inequality.
The appendix proves a trace inequality on polygonal or polyhedral elements by mapping each face through an interior pyramid and integrating along face-to-apex segments.
- Trace inequality: Lemma A.1 establishes a trace inequality for θ ∈ W^(1,p)(T) on every element face when p > 1.The mesh must satisfy assumptions A1, A2, and A3.
- Geometric construction: The proof parameterizes a flat face and constructs an interior pyramid with the face as base and an element point as apex.The face is represented by xe = φ(ξ, η), and the pyramid lies inside T.
- Proof mechanism: Values along each face-to-apex segment are related using the fundamental theorem of calculus.This converts face information into volume estimates over the pyramid or prismatoid.
- Integral estimate: Cauchy–Schwarz and an auxiliary inequality yield the face integral bound after integration over the face and prismatoid.The remaining integrations establish the stated trace inequality and complete the proof.
- Jacobian control: The coordinate transformation uses a Jacobian determined by the face normal and the vector from the face point to the apex.The angle assumption bounds the Jacobian through a fixed parameter μ0 ∈ (0, 1).
A.3. A Domain Inverse Inequality.
This section develops domain inverse inequalities, first for functions and polynomials on convex domains, then for piecewise polynomials on shape-regular polytopal partitions.
- Function estimates: Lemma A.2 establishes an estimate for functions v in W 1,p(K) on a convex domain K.The proof begins with smooth functions and uses density of C1(K) in W 1,p(K).
- Geometric assumptions: Shape regularity supplies volume scaling and an inscribed ball, supporting the comparison between norms on K and on an interior ball S.The relevant geometric properties are that the measure of K scales with hK^d and that K contains an inscribed ball of comparable diameter.
- Polynomial estimates: Recursive application of the estimate for smooth functions yields a corresponding bound for polynomial functions.The polynomial case is combined with the standard inverse inequality.
- Simplex domain inequality: Lemma A.3 gives a domain inverse inequality for degree-n polynomials on a shape-regular d-simplex K using a ball S whose diameter is proportional to hK.Its constant depends on the proportionality parameter and polynomial degree.
- Polytopal partitions: Lemma A.4 extends the inverse inequality to piecewise polynomials of degree n on polygonal or polyhedral partitions satisfying assumptions A1-A4.The proof combines Lemma A.3 with the standard inverse inequality on shape-regular circumscribed simplices.
- Further norm comparison: Lemma A.5 provides a further inverse estimate for piecewise polynomials when two exponents satisfy p ≥ r ≥ 1.The result applies on the same class of polytopal partitions under assumptions A1-A4.