PaperlayerView exact revision on arXiv

Source-linked AI summary

A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form

Chunmei Wang, Junping Wang

arXiv:1510.03499v1math.NAcs.NA

TL;DR

For non-smooth coefficients in L∞(Ω), a variational formulation using integration by parts is not possible for the non-divergence-form problem. The method uses a locally reconstructed discrete weak Hessian and minimizes a non-negative quadratic functional subject to the discrete equation, producing a primal-dual scheme. Under H2 regularity and piecewise-continuous uniformly elliptic coefficients, the paper derives optimal-order error estimates in the discrete H2 norm for the primal variable and the L2 norm for the dual variable.

  • Problem

    For non-smooth coefficients in L∞(Ω), a variational formulation using integration by parts is not possible for the non-divergence-form problem.

    Source: §1 · pp. 1–2

  • Method

    The method uses a locally reconstructed discrete weak Hessian and minimizes a non-negative quadratic functional subject to the discrete equation, producing a primal-dual scheme.

    Source: §1 · pp. 2–3

  • Results

    Under H2 regularity and piecewise-continuous uniformly elliptic coefficients, the paper derives optimal-order error estimates in the discrete H2 norm for the primal variable and the L2 norm for the dual variable.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Numerical experiments with smooth and non-smooth coefficients on convex and non-convex domains illustrate accuracy and confirm the developed theory.

    Source: §1 · p. 3

    The optimal-order error estimate (7.24) assumes P1(T) ⊆ S2(T); otherwise inequalities (7.26), (7.28), and (7.29) need modification.

    Source: §7 · p. 22

Abstract

from arXiv · show

This article proposes a new numerical algorithm for second order elliptic equations in non-divergence form. The new method is based on a discrete weak Hessian operator locally constructed by following the weak Galerkin strategy. The numerical solution is characterized as a minimization of a non-negative quadratic functional with constraints that mimic the second order elliptic equation by using the discrete weak Hessian. The resulting Euler-Lagrange equation offers a symmetric finite element scheme involving both the primal and a dual variable known as the Lagrange multiplier, and thus the name of primal-dual weak Galerkin finite element method. Optimal order error estimates are derived for the finite element approximations in a discrete $H^2$-norm, as well as the usual $H^1$- and $L^2$-norms. Some numerical results are presented for smooth and non-smooth coefficients on convex and non-convex domains.

1. Introduction.

The paper develops a primal-dual weak Galerkin method for non-divergence-form elliptic problems, motivated by rough coefficients that prevent a variational formulation using integration by parts. It uses a discrete weak Hessian and constrained minimization, and derives error estimates under stated regularity and coefficient assumptions.

1. Introduction.

  • For non-smooth coefficients a(x) in L∞(Ω), integration by parts does not yield a variational formulation, motivating methods for non-divergence-form problems.The paper notes applications involving essentially bounded coefficients in Hamilton-Jacobi-Bellman equations and piecewise-smooth coefficients from linearized fully nonlinear PDEs.Source: §1 · p. 2
  • Earlier approaches include conforming finite elements, an interior-penalty primal method, and an hp-discontinuous Galerkin least-squares method with optimal mesh-size but half-order-suboptimal polynomial-degree convergence,, [9],.The cited hp method applies to problems satisfying the Cordès condition.Source: §1 · p. 2
  • The method reconstructs the Hessian locally as a discrete weak Hessian and uses it to approximate the model problem.The weak Galerkin strategy uses locally constructed differential operators on discontinuous functions that include boundary information.Source: §1 · p. 2
  • A non-negative quadratic continuity functional is minimized subject to the discrete equation, yielding a symmetric scheme with primal and Lagrange-multiplier variables.The direct discrete problem is not well-posed unless a Babuška–Brezzi inf-sup condition holds; the optimization formulation is proposed to overcome this difficulty.Source: §1 · pp. 2–3
  • With an H2-regular solution and piecewise-continuous uniformly elliptic coefficients, the paper derives optimal-order discrete H2 estimates for the primal variable and L2 estimates for the dual variable.H1- and L2-norm convergence estimates for the primal variable require additional coefficient smoothness; experiments cover smooth and non-smooth coefficients on convex and non-convex domains.Source: §1 · p. 3

2. Preliminaries.

The preliminaries state regularity and coefficient conditions for strong solutions and set up a variational equation. They also describe the Cordès condition, a uniqueness concern for discontinuous coefficients in dimensions three and higher, and the weak-function framework.

2. Preliminaries.

  • Classical Schauder and Calderón–Zygmund results provide strong-solution regularity under their respective coefficient and boundary smoothness conditions.The stated conditions are C0,α coefficients with a C2,α boundary, or continuous coefficients with a C1,1 boundary.Source: §2 · p. 3
  • For discontinuous coefficients in dimensions d ≥ 3, solution uniqueness may fail; the paper therefore assumes a Cordès condition in that setting.For bounded convex domains, the stated ellipticity and Cordès conditions give a unique H2(Ω) ∩ H1_0(Ω) strong solution for each f in L2(Ω).Source: §2 · p. 4
  • The analysis assumes a unique strong solution in H2(Ω) ∩ H1_0(Ω) with an a priori estimate, while noting a W2,p alternative.The variational formulation can use X = W2,p(Ω) ∩ W1,p_0(Ω) and Y = Lq(Ω) when the corresponding W2,p regularity holds.Source: §2 · pp. 4–5
  • The strong problem is expressed as b(u,w) = (f,w) for all w in Y, with b(v,σ) = (Lv,σ); the regularity assumption supplies an inf-sup condition.The text identifies X with the H2 norm and Y with the L2 norm.Source: §2 · p. 5
  • A weak function on an element is a triplet of interior value, boundary value, and boundary gradient components, which need not be related by trace conditions.When the boundary components are traces and exist, the weak function is determined by its interior component and is classical.Source: §2 · p. 5

3. Weak Hessian and Discrete Weak Hessian.

The section motivates a Hessian-focused discretization because the Hessian is the primary differential operator in the non-divergence-form problem. It reviews a weak Hessian and its discrete counterpart for weak functions.

3. Weak Hessian and Discrete Weak Hessian.

  • The Hessian is the primary differential operator in the non-divergence-form problem, so the section targets its numerical reconstruction.The section reviews the discrete weak Hessian introduced in [19].Source: §3 · p. 5
  • A weak function on K has interior and boundary components, and its generalized weak second derivative is defined as a bounded linear functional on H2(K).The functional's action is specified for test functions in H2(K), with the boundary normal appearing in the definition.Source: §3 · pp. 5–6
  • The discrete weak Hessian is formed by a polynomial-space analogue of the generalized weak second derivative.For a smooth interior component, integration by parts can be applied to the first term in the weak-derivative definition.Source: §3 · p. 6

4. Primal-Dual Weak Galerkin.

The method is built on polynomial weak finite element spaces over polygonal or polyhedral meshes and a discrete weak Hessian. It formulates the discrete PDE as constrained optimization and obtains the primal-dual scheme from its Euler–Lagrange equations.

4. Primal-Dual Weak Galerkin.

  • The method uses shape-regular polygonal meshes in 2D or polyhedral meshes in 3D, with elementwise polynomial weak functions and shared interface components.The local space uses degree-k interior and boundary values and degree-(k−1) boundary gradients.Source: §4 · p. 6
  • The Lagrange-multiplier space is chosen between Pk−2(T) and Pk−1(T), with the lower-degree choice using fewer degrees of freedom but potentially reducing accuracy.The discrete weak second-order operator uses the selected multiplier polynomial space.Source: §4 · pp. 6–7
  • Using the discrete bilinear forms, the method casts the elliptic problem as constrained optimization and derives its numerical scheme from the Euler–Lagrange equation.The scheme seeks a primal approximation and a dual variable.Source: §4 · p. 7

5. Stability and Solvability.

The section establishes stability and solvability for the primal-dual weak Galerkin scheme using norm equivalence and saddle-point conditions. Under piecewise-continuous coefficients and a sufficiently small mesh size, the scheme has a unique solution.

5. Stability and Solvability.

  • The discrete norm and a related seminorm are equivalent under the stated piecewise-continuity condition.The two differ by replacing strong second derivatives with discrete weak second derivatives in one term.Source: §5 · pp. 9–10
  • An inf-sup condition is established when the coefficient matrix is uniformly piecewise continuous and the mesh size is sufficiently small.The proof constructs a test function using a projection of a solution to an auxiliary problem.Source: §5 · pp. 10, 12
  • The stability analysis establishes boundedness and coercivity on the relevant subspace.The coercivity argument uses the discrete norm equivalence.Source: §5 · pp. 12–13
  • For sufficiently small mesh size, the primal-dual weak Galerkin scheme has a unique solution under the stated coefficient condition.The theorem also gives bounds for the primal and dual variables.Source: §5 · p. 13

6. Error Estimates.

This section derives error equations for the primal-dual scheme and uses stability of its saddle-point system to obtain an error estimate. The estimate assumes piecewise-continuous coefficients, a sufficiently regular exact solution, and a sufficiently small mesh size.

6. Error Estimates.

  • The primal and dual errors satisfy a pair of equations that form a saddle-point system for stability analysis.The dual error equals the discrete Lagrange multiplier because the continuous dual solution is zero.Source: §6 · pp. 13–14
  • Theorem 6.3 gives an error estimate for the primal-dual scheme when coefficients are uniformly piecewise continuous and the exact solution satisfies the stated regularity condition.The proof invokes the stability conditions for the saddle-point system and bounds terms involving the stabilization form.Source: §6 · pp. 15–16

7. Error Estimates in H1 and L2.

The section derives H1- and L2-norm error estimates for the primal variable using dual-problem regularity. It also gives boundary-component estimates and identifies a polynomial-space condition needed for the optimal-order L2 estimate.

7. Error Estimates in H1 and L2.

  • Under the stated coefficient smoothness, solution regularity, and dual H1-regularity assumptions, the analysis establishes an H1 error estimate for the primal variable.The proof uses a dual problem with data related to the error and combines the resulting bound with the discrete H2 estimate.Source: §7 · pp. 19, 21
  • The L2 error estimate assumes coefficient smoothness, dual H2-regularity, and P1(T) ⊂ S_k(T), with sufficiently small mesh size.The proof combines the preceding error estimates with bounds obtained from the dual problem.Source: §7 · pp. 21–22
  • If P1(T) is not contained in S2(T), the stated optimal-order L2 estimate is replaced by a sub-optimal-order estimate.The modification replaces a bound involving the dual solution's H2 norm with one involving its H1 norm and h^-1.Source: §7 · p. 22
  • The analysis also provides error estimates for the boundary components ub and ug under the assumptions of Theorem 7.6.The estimates follow from trace bounds and the previously derived error estimates.Source: §7 · pp. 22–23

8. Numerical Results.

The experiments test the primal-dual WG method with C0-type elements on triangular meshes, across continuous and discontinuous coefficients and convex and non-convex domains. The reported rates generally agree with theory for gradient and multiplier approximations, while some solution-variable rates exceed available predictions.

8. Numerical Results.

  • The experiments use lowest-order WG elements on triangular partitions, including C0-type variants that reduce linear-system complexity but restrict the available polygonal elements.The mesh sequence comes from uniform refinement, and the weak second derivatives for C0-type elements are computed in the local space S2(T).Source: §8 · pp. 23–24

8.1. Numerical experiments with continuous coefficients.

  • For smooth constant coefficients, u0 converges at r = 4.0 on the unit square and r = 3.5 on the L-shaped domain; the discrete H1-seminorm rate is r = 2.0 on both.The Lagrange multiplier rate is r = 1.0 on both domains; the authors report that u0's L2 convergence exceeds theory and suggest superconvergence for smooth data on uniform triangular partitions.Source: §8.1 · p. 25, Table 8.1 · p. 25, Table 8.2 · p. 25
  • With varying continuous coefficients, performance is similar to the constant-coefficient case, but the apparent superconvergence weakens.The coefficient is continuous, but its derivative is singular at the origin, so the equation cannot be written in divergence form.Source: §8.1 · p. 25, Table 8.3 · p. 26
  • With the P0 multiplier element, the usual L2 convergence rate is r = 2.0 for both the gradient approximation and u; the multiplier converges slightly faster than r = 1.0.The primal-variable convergence agrees with theory, while the dual-variable convergence exceeds the theoretical rate.Source: §8.1 · p. 26, Table 8.4 · p. 26

8.2. Numerical experiments with discontinuous coefficients.

  • For discontinuous coefficients in problem (8.2), the gradient approximation converges at r = 2.0 in L2, while u's L2 rate appears higher than r = 2.The multiplier rate also appears higher than the predicted r = 1.0; the authors note that optimal-order L2 error estimates for u lack theoretical support because the needed dual-problem regularity is unclear.Source: §8.2 · p. 27, Table 8.5 · p. 27
  • For problem (8.2), the P0-multiplier element gives smaller absolute errors than the P1-multiplier element, with comparable convergence rates.Source: §8.2 · p. 27, Table 8.6 · p. 27, Table 8.5 · p. 27
  • For problem (8.4) on the unit square, the multiplier's L2 rate is r = 0.6 and the H1-seminorm rate is r = 1.6; u's L2 rate appears to be r = 2.On the reference square, where the coefficient is discontinuous at the center, results are less accurate; the duality argument does not apply, while H1 and L2 rates appear to be r = 1.0 or slightly higher.Source: §8.2 · pp. 28–29, Table 8.7 · p. 28, Table 8.9 · p. 29
Loading 1510.03499v1…