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A Computational Study of the Weak Galerkin Method for Second-Order Elliptic Equations
Lin Mu, Junping Wang, Yanqiu Wang, Xiu Ye
TL;DR
The paper examines computation and numerical accuracy for the weak Galerkin method on second-order elliptic problems across broader finite-element partitions. Its numerical experiments confirm prior theoretical predictions and also report behavior that remains without a theoretical foundation.
Problem
Method
The study analyzes the method on multiple mesh types and tests it on model problems, including cases with varied boundary conditions and degenerate diffusion.
Source: §2 · p. 2, §4.1 · p. 12, §4.2 · p. 13Results
Takeaways & Limitations
The experiments report optimal convergence rates for mixed boundary conditions and both tested boundary-data approximations, while also documenting convergence for a degenerate-diffusion case.
Source: §4.1 · p. 12, §4.2 · p. 13A theoretical foundation for the observed convergence behavior in the degenerate-diffusion case is left for future research.
Source: §4.2 · p. 14
Abstract
from arXiv · showhide
The weak Galerkin finite element method is a novel numerical method that was first proposed and analyzed by Wang and Ye for general second order elliptic problems on triangular meshes. The goal of this paper is to conduct a computational investigation for the weak Galerkin method for various model problems with more general finite element partitions. The numerical results confirm the theory established by Wang and Ye. The results also indicate that the weak Galerkin method is efficient, robust, and reliable in scientific computing.
1. Introduction.
The paper investigates computation and numerical accuracy for the weak Galerkin method on second-order elliptic problems. It extends the study across several mesh types and reports numerical results that confirm prior theory while revealing results not yet theoretically proved.
1. Introduction.
- The study examines computation and numerical accuracy for the weak Galerkin method applied to a linear second-order elliptic boundary-value problem.The model is posed on a bounded Lipschitz domain in two or three dimensions, with a unique weak solution assumed.Source: §1 · p. 1
- The numerical experiments confirm predictions from the original theory [23] and reveal results not yet theoretically proved.The paper also introduces the method and abstract theory, proves rectangular elements satisfy its assumptions, and reports test problems.Source: §1 · p. 2
- Meshes considered include triangles and rectangles in two dimensions, and mainly tetrahedra and hexahedra in three dimensions.The mesh is assumed shape-regular and quasi-uniform.Source: §2 · p. 2
2. The Weak Galerkin Method.
The weak Galerkin method represents element-interior and boundary values separately and uses a discrete gradient in its variational formulation. The paper establishes the conditions behind the method’s approximation properties and extends the analysis to rectangular elements.
2. The Weak Galerkin Method.
- The weak discrete space Sh permits discontinuities across element and edge or face interfaces, representing each function by interior and boundary components, v = {v0, vb}.The method uses an L2 projection into Sh and a locally defined discrete gradient.Source: §2 · p. 2
- The discrete formulation seeks uh with prescribed projected boundary values and uses a bilinear form combining diffusion, advection, and reaction terms.The spaces must meet conditions that preserve information under discrete gradients while keeping computational cost sufficiently small.Source: §2 · p. 3
- Under P1 and P2, prior analysis [23] gives unique solvability for moderately small mesh size when the dual problem has H1+s regularity for some s > 0, together with an error estimate.The largest admissible s is determined by the dual problem’s H1+s regularity.Source: §2 · p. 3
2.1. Weak Galerkin on Rectangular Meshes.
- The rectangular choices are (Qj(K0), Qj(F), RTj(K)) and (Pj(K0), Pj+1(F), BDMj+1(K)); a three-dimensional extension to tetrahedral and hexahedral elements is described as straightforward.For rectangles, the two choices satisfy P1 and P2.Source: §2 · p. 3, §2.1 · pp. 4–5
- For rectangular elements, the paper verifies assumptions P1 and P2 and derives the corresponding error estimate by following the argument in [23].The discrete gradient of the projected function is the L2 projection of its gradient onto RTj(K) or BDMj+1(K).Source: §2.1 · pp. 4–5
- Implementation reduces the method to a matrix assembled from element-local stiffness matrices, whose computation is the section’s focus.The paper presents ways to compute these matrices for elements introduced earlier.Source: §2.1 · p. 5
3. Computation of Local Stiffness Matrices.
The section gives a matrix-based procedure for computing local stiffness matrices, first for a lowest-order triangular element and then for a cubic element. Two triangular-element basis approaches yield the same local stiffness matrix, while the second makes the inverse of DK trivial to compute.
3. Computation of Local Stiffness Matrices.
- The local stiffness matrix MK is computed from AK, BK, and CK using Lemma 3.1, with DK, ZK, and TK used to compute weak gradients of basis functions.For the Poisson equation, AK = DK and BK = CK = 0.Source: §3 · p. 7
3.1. For the Triangular Element (P0(K), P0(F), RT0(K).
- For triangular elements, the example uses piecewise constants on triangles and edges with the lowest-order Raviart-Thomas element for the discrete gradient.The local stiffness matrix is represented in blocks corresponding to interior and edge degrees of freedom.Source: §3 · pp. 6–7, §3.1 · p. 8
3.1.1. Approach I.
- Approach I expresses a local matrix form using only edge lengths, which the authors identify as convenient for computer implementation.The paper notes that DK agrees with the formula presented in.Source: §3.1.1 · pp. 8–9
3.1.2. Approach II.
- Approach II uses an alternative basis for Vr(K) and the same weak-discrete-function basis, producing the same MK as Approach I.Its DK is diagonal, so its inverse is trivial to compute.Source: §3.1.2 · pp. 9–10
3.2. For the Cubic Element (Q0(K), Q0(F), RT0(K)).
- For a cubic element, the discrete space uses piecewise constants on the element and its faces, with the lowest-order Raviart-Thomas element for the discrete gradient.The local stiffness matrix is computed using the formula in Lemma 3.1.Source: §3.1.2 · p. 10, §3.2 · p. 11
4. Numerical Experiments.
The numerical experiments test weak Galerkin approximations across boundary conditions, degenerate diffusion, singular solutions, interfaces, anisotropic meshes, and a three-dimensional problem. The results include optimal convergence in several settings, slower observed rates for degenerate diffusion, and superconvergence in one three-dimensional metric.
4. Numerical Experiments.
- The experiments define the approximation error as eh = uh − Qhu and evaluate it with several norms and metrics, using high-order Gaussian quadrature for the integrals.The maximum norm ∥e0∥∞ is computed over Gaussian points.Source: §4 · p. 11
4.1. Case 1: Model Problems with Various Boundary Conditions.
- Both nodal interpolation and L2 projection of Dirichlet data yielded optimal convergence; L2 projection gave slightly smaller ∥e0∥ and ∥eb∥ errors, and the mixed-boundary test also showed optimal rates in all norms.The mixed-boundary test imposes a Robin condition on part of the boundary.Source: §4.1 · p. 12, Table 4.1 · p. 13, Table 4.2 · p. 13
4.2. Case 2: A Model Problem with Degenerate Diffusion.
- Degenerate diffusion produced rates of approximately O(h^0.5) in ∥∇deh∥ and O(h^1.25) in ∥e0∥ and ∥eb∥, while the theory for this behavior remains open.The coefficient is not uniformly positive definite, and the authors leave a theoretical foundation for the observed rates to future research.Source: §4.2 · p. 13, Table 4.4 · p. 14, Figure 4.1 · p. 15
4.3. Case 3: A Model Problem on a Domain with Corner Singularity.
- For corner singularities tested at γ = 0.5 and γ = 0.25, ∥∇deh∥, ∥e0∥, and ∥eb∥ converged at rates O(h^γ), O(h^(1+γ)), and O(h^(1+γ)), respectively.The reported rates for ∥∇deh∥ and ∥e0∥ agree with the theoretical predictions.Source: §4.3 · p. 14, Table 4.5 · p. 15, Table 4.6 · p. 16
4.4. Case 4: A Model Problem with Intersecting Interfaces.
- For the intersecting-interface problem, locally refined initial meshes improved accuracy, and refining more near the origin slightly increased the observed convergence rates.Because the mesh is not quasi-uniform, the authors do not expect the theoretical error estimate to apply; they also report that ∥u0 − u∥ initially converges faster than ∥e0∥.Source: §4.4 · p. 16, Table 4.7 · p. 17, Table 4.8 · p. 18
4.5. Case 5: An Anisotropic Problem.
- For the anisotropic problem, tests with k = 3 and k = 9 showed optimal convergence rates in various metrics, indicating that the method handled the tested anisotropic problems and meshes.The meshes were formed by splitting sub-rectangles into triangles.Source: §4.5 · p. 17, Table 4.9 · p. 18, Table 4.10 · p. 18
4.6. Case 6: A Three-Dimensional Model Problem.
- For the three-dimensional Laplace test, the cubic element achieved optimal convergence rates and superconvergence in ∥∇deh∥.The authors identify further investigation, especially for variable-coefficient problems, as an open direction.Source: §4.6 · p. 17, Table 4.11 · p. 18