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An Efficient Numerical Scheme for the Biharmonic Equation by Weak Galerkin Finite Element Methods on Polygonal or Polyhedral Meshes

Chunmei Wang, Junping Wang

arXiv:1309.5560v1math.NAcs.NA

TL;DR

Constructing practical H2-conforming elements for the biharmonic equation is difficult because they require C1 continuity. This paper develops a weak Galerkin method using discrete weak derivatives and lower-degree boundary polynomials; it establishes error estimates and reports exact numerical agreement for a quadratic solution.

  • Problem

    H2-conforming finite element methods for the biharmonic equation require C1-continuous piecewise polynomials, whose construction is difficult and rarely used in practice.

    Source: §1 · p. 2

  • Method

    The method uses weak Galerkin elements with degree-k interior polynomials, degree-(k−2) boundary polynomials, and discrete weak derivatives.

    Source: §1 · p. 2

  • Results

    The paper establishes an error estimate in an H2-equivalent norm and an L2 error estimate under stated regularity assumptions, and reports numerical agreement with the exact solution for a quadratic polynomial.

    Source: §7 · p. 12, §8 · p. 13, §9 · p. 19

  • Takeaways & Limitations

    The proposed WG element type reduces degrees of freedom and the system size relative to the earlier WG method [13].

    Source: §1 · p. 3

    The authors call for more experiments, particularly for orders above k=2 and arbitrary polygonal partitions, and for fast solvers for the resulting matrix problem.

    Source: §9 · p. 20

Abstract

from arXiv · show

This paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak Galerkin (WG) finite element methods. The WG finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual $H^2$-semi norm. Weak partial derivatives and their approximations, called discrete weak partial derivatives, are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. The discrete weak partial derivatives serve as building blocks for the WG finite element method. The resulting matrix from the WG method is symmetric, positive definite, and parameter free. An error estimate of optimal order is derived in an $H^2$-equivalent norm for the WG finite element solutions. Error estimates in the usual $L^2$ norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.

1. Introduction.

The paper addresses the difficulty of constructing practical H2-conforming finite elements for the biharmonic equation and develops a weak Galerkin alternative. Its lower-degree boundary components reduce the number of unknowns relative to an earlier WG method, and the paper establishes convergence estimates.

1. Introduction.

  • Constructing H2-conforming finite elements is difficult because their piecewise polynomials require C1 continuity, so these methods are rarely used in practice for the biharmonic equation.The introduction also notes alternative nonconforming, discontinuous Galerkin, and mixed approaches to avoid C1 elements.Source: §1 · p. 2
  • The proposed WG method uses degree-k polynomials inside elements and degree-(k−2) polynomials on element boundaries for the function and its gradient.It is based on the biharmonic variational form and replaces second derivatives with discrete weak versions, with a stabilizer enforcing weak continuity.Source: §1 · p. 2
  • Compared with the earlier WG method [13], the proposed element type reduces degrees of freedom and yields a smaller system to solve.The paper states that it develops a mathematical convergence theory for the method.Source: §1 · p. 3

2. Preliminaries and Notations.

This section establishes notation for Sobolev spaces and norms used throughout the analysis.

2. Preliminaries and Notations.

  • For bounded Lipschitz domains in dimensions 2 or 3, the paper uses standard Sobolev spaces H^s(D), their inner products, norms, and seminorms.For integer s, the seminorm is defined using derivatives indexed by multi-indices.Source: §2 · p. 3
  • The space H^0(D) is identified with L2(D), whose norm and inner product are denoted by ∥·∥D and (·,·)D.When D=Ω, the domain subscript is omitted.Source: §2 · p. 3
  • Throughout the paper, C denotes a generic constant independent of the mesh size and the functions involved.Source: §2 · p. 3

3. Weak Partial Derivatives of Second Order.

The section defines weak second partial derivatives for discontinuous functions on polygonal or polyhedral elements and introduces polynomial approximations for computation.

3. Weak Partial Derivatives of Second Order.

  • A weak function on an element is a triple {v0, vb, vg}, with interior value v0 and boundary components vb and vg representing the function and its gradient.The boundary components need not equal traces of the interior function and gradient.Source: §3 · p. 4
  • The weak second partial derivative is defined as a linear functional acting on test functions, extending the classical second derivative to weak functions.For functions in H2(T), the weak derivative agrees with the classical derivative.Source: §3 · p. 4
  • For computation, the weak second derivative is discretized in polynomial subspaces, using test polynomials of degree at most r.The section introduces this discrete operator after defining the weak derivative.Source: §3 · p. 4

4. Numerical Algorithm by Weak Galerkin.

The WG algorithm combines discrete weak second derivatives with a stabilizer on polygonal or polyhedral meshes. The paper establishes uniqueness of its discrete solution.

4. Numerical Algorithm by Weak Galerkin.

  • The mesh partitions Ω into polygons in 2D or polyhedra in 3D and is assumed to satisfy shape regularity.Source: §4 · p. 5
  • The discrete element space uses degree-k interior polynomials and degree-(k−2) boundary polynomials for the function and its gradient, patched across interior faces.Boundary values vanish on boundary faces in the subspace V0_h.Source: §4 · p. 5
  • The algorithm imposes projected boundary data and solves a variational equation combining discrete weak second derivatives and a stabilizer against the forcing term.The discrete weak derivative is computed with polynomial degree k−2, and the stabilizer supports weak continuity.Source: §4 · pp. 5–6
  • The discrete energy expression defines a norm on V0_h, and the WG algorithm has a unique solution.The uniqueness result follows from the norm property.Source: §4 · p. 6

5. L2 Projections and Their Properties.

The section establishes projection properties used in the WG error analysis, including a commutative relation between L2 projections and discrete weak derivatives. It also presents approximation results under stated regularity assumptions.

5. L2 Projections and Their Properties.

  • The section develops technical results for L2 projections because they are valuable in the WG method’s error analysis.These results include projection approximation properties.Source: §5 · p. 7
  • The discrete weak derivative of the L2 projection of a smooth function approximates that function’s classical derivative.The commutative property is useful in algorithm design and analysis.Source: §5 · p. 7
  • Lemma 5.3 states approximation estimates for 1 ≤ m ≤ k when u ∈ H^max{m+1,4}(Ω).The passage identifies δ_i,j as the Kronecker delta.Source: §5 · pp. 7–8

6. An Error Equation.

The section derives an equation satisfied by the error between the projected exact solution and its WG approximation. The derivation combines integration by parts with the WG equation and boundary cancellation.

6. An Error Equation.

  • The error function e_h, defined from the L2 projection of u and its WG approximation, belongs to V_h^0 and satisfies an error equation.The equation is obtained by subtracting the WG scheme after forming the corresponding identity for the projected exact solution.Source: §6 · pp. 9–10
  • The derivation sums elementwise identities, with terms associated with v_g·n_j and v_b n_i vanishing on the boundary.The argument uses integration by parts and the identity (∆²u, v_0) = (f, v_0).Source: §6 · pp. 9–10
  • Adding the stabilizer term s(Q_hu,v) before subtracting the WG equation yields the error equation.This links the projected exact solution to the discrete WG solution.Source: §6 · p. 10

7. Error Estimates in H2.

The section proves an optimal-order error estimate in an H2-equivalent discrete norm for WG elements of order k ≥ 2, assuming sufficient solution regularity. It bounds the error by estimating terms in the error equation.

7. Error Estimates in H2.

  • The error estimate is obtained by controlling φ_u(v), the term identified as sufficient to handle in the error equation.The section bounds the component terms I_j(w,v) using projection estimates and separate treatment for k = 2 and k ≥ 3.Source: §7 · pp. 10–11
  • For the I2 term, the bound is |I2(w,v)| ≤ C h^(r−1)(∥w∥_(r+1) + δ_(k,2)∥w∥_4) |||v|||.The δ_(k,2) term captures an additional contribution for quadratic elements.Source: §7 · p. 11
  • For k ≥ 2 and u ∈ H^max{k+1,4}(Ω), the WG error has optimal-order convergence in the H2-equivalent trip-bar norm.The estimate follows by testing the error equation with the error function and bounding its component terms.Source: §7 · p. 12

8. Error Estimates in L2.

The section estimates the WG solution’s error in the usual L2 norm using a dual problem. The resulting convergence order is suboptimal for k = 2 and optimal for k ≥ 3, under solution and dual regularity assumptions.

8. Error Estimates in L2.

  • The L2 estimate is derived for the first component e_0 of the error function by considering a dual problem.The error identity splits ∥e_0∥_2^2 into terms involving φ_u(Q_hψ) and φ_ψ(e_h).Source: §8 · pp. 12–13
  • L2 convergence is suboptimal for k = 2 and optimal for k ≥ 3, assuming the stated solution regularity and H4 regularity of the dual problem.The theorem sets t_0 = min{k, 3}.Source: §8 · p. 13
  • The analysis bounds φ_u(Q_hψ) using estimates for I3 and I4 and the bound |Q_hψ|_h ≤ C h^(t_0−1)∥e_0∥.It also uses the orthogonality of Q_b and the zero boundary value of ψ to estimate I1 and I2.Source: §8 · pp. 14–15
  • The H2 and L2 error estimates are used to derive estimates for the WG solution components u_b and u_g.The analysis represents their errors through e_0 and the discrete weak derivative of e_h.Source: §8 · p. 15

9. Numerical Experiments.

The experiments test the lowest-order Weak Galerkin scheme on triangular and rectangular meshes, including polynomial and trigonometric exact solutions. Reported results include the expected H2 and L2 convergence rates and exact recovery of quadratic polynomials; the authors identify further experiments and faster solvers as needed.

9. Numerical Experiments

  • The experiments implement the lowest-order scheme on the unit square using uniform triangular and rectangular partitions, with mesh size h = 1/n.The authors call for more experiments, especially for orders above k = 2 and arbitrary polygonal partitions, and for development of faster solvers.Source: §9 · pp. 15, 16, 20
  • For a quadratic exact solution on triangular partitions, the method is theoretically exact, and Table 9.1's computational results are reported as consistent with that property.The authors describe this test as indicating that the code is working.Source: §9 · p. 16, Table 9.1 · p. 16
  • For the homogeneous-boundary test on triangular meshes, the Weak Galerkin algorithm converges at O(h) in H2 and O(h^2) in L2.The authors report that these results are consistent with theory.Source: §9 · pp. 16–17
  • For the triangular-mesh test, the edge-set L∞ error for u appears to be O(h^2), but the convergence order for the edge approximation of ∇u is difficult to determine from the data.The absolute errors for both ub and ug are described as quite small.Source: §9 · p. 17
  • For the biquadratic-polynomial test on triangular meshes, L2 convergence is O(h^2), while H2 convergence is approximately O(h).The table caption identifies triangular meshes; the reported rates are stated in the accompanying text.Source: §9 · p. 18, Table 9.6 · p. 18
  • On rectangular meshes, the quadratic-polynomial test produces a numerical solution that is numerically the same as the exact solution, as predicted by theory.The lowest-order rectangular element has 18 unknowns per element, reduced to 12 by local elimination of the unknowns for u.Source: §9 · pp. 18–19, Table 9.7 · p. 19

10. Appendix.

The appendix develops foundational estimates used in error analysis for general weak Galerkin finite element methods. It also establishes supporting identities and inequalities, including Poincaré estimates and a regularity condition for a quadratic-element estimate.

10. Appendix.

  • The appendix develops foundational estimates used in error analysis for general WG finite element methods.Source: §10 · p. 20
  • The appendix invokes trace and inverse inequalities, including an inverse inequality for polynomial functions on an element.Source: §10 · p. 20
  • An elementwise identity for weak partial derivatives follows from their definition and integration by parts.Source: §10 · pp. 20–21
  • A Poincaré inequality is established for finite element functions in V_h with e_b = 0.Source: §10 · p. 22
  • For quadratic elements, Lemma 10.6 assumes the exact solution lies in H^4(Ω) to establish its estimate.Source: §10 · p. 23
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