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A Weak Galerkin Finite Element Method for Second-Order Elliptic Problems
Junping Wang, Xiu Ye
TL;DR
The paper addresses how to construct finite element approximations for second-order elliptic problems without requiring the continuity typically used to apply classical gradients. It defines weak and discrete weak gradients, builds a weak Galerkin method from them, and reports well-posedness, optimal-order error estimates, and superconvergence.
Problem
Classical-gradient Galerkin methods typically require continuous piecewise-polynomial approximations, while existing discontinuous methods still interpret the gradient classically.
Method
The paper defines weak gradients for functions with separate interior and boundary components, approximates them in polynomial H(div) subspaces, and uses them in a weak Galerkin method.
Results
The method has a unique solution under an H1+s dual-regularity assumption for sufficiently small mesh size and exhibits optimal-order error estimates and superconvergence.
Takeaways & Limitations
Weak Galerkin finite elements provide a framework for using totally discontinuous functions in the finite element procedure.
Abstract
from arXiv · showhide
In this paper, authors shall introduce a finite element method by using a weakly defined gradient operator over discontinuous functions with heterogeneous properties. The use of weak gradients and their approximations results in a new concept called {\em discrete weak gradients} which is expected to play important roles in numerical methods for partial differential equations. This article intends to provide a general framework for operating differential operators on functions with heterogeneous properties. As a demonstrative example, the discrete weak gradient operator is employed as a building block to approximate the solution of a model second order elliptic problem, in which the classical gradient operator is replaced by the discrete weak gradient. The resulting numerical approximation is called a weak Galerkin (WG) finite element solution. It can be seen that the weak Galerkin method allows the use of totally discontinuous functions in the finite element procedure. For the second order elliptic problem, an optimal order error estimate in both a discrete $H^1$ and $L^2$ norms are established for the corresponding weak Galerkin finite element solutions. A superconvergence is also observed for the weak Galerkin approximation.
1. Introduction.
The paper introduces weak and discrete weak gradients to approximate second-order elliptic problems while allowing discontinuous functions with separate interior and boundary values.
- Existing Galerkin methods typically use continuous piecewise polynomials so the classical gradient can be applied.
- The classical gradient remains standard even in existing discontinuous Galerkin methods, motivating a different interpretation of differential operators.
- Replacing the classical gradient with a discrete weak gradient yields the weak Galerkin finite element method.
- The weak gradient is defined on heterogeneous functions whose interior and boundary components may be unrelated.
- Its action is a bounded linear functional on H(div, K), making the weak gradient well defined.
- The method is closely related to hybridized mixed finite elements and introduces a systematic framework for discontinuous element and boundary functions.
2. Preliminaries and Notations.
The preliminaries establish standard Sobolev-space notation and define H(div; Ω) for square-integrable vector fields whose divergence is also square integrable.
- Sobolev spaces H^s(D) are equipped with their standard inner products, norms, and seminorms.
- For integer s ≥0, Sobolev seminorms are expressed using multi-index derivatives.
- H(div; Ω) consists of vector-valued functions on Ω whose components and divergence are square integrable.
3. A Weak Gradient Operator and Its Approximation.
The paper defines weak gradients as dual-space functionals and approximates them in polynomial H(div) subspaces, enabling differentiation without inter-element continuity.
- A weak function on K stores an interior value v0 and a boundary value vb, with vb not necessarily the trace of v0.
- The weak gradient ∇dv is defined as a linear functional in the dual of H(div, K).
- The weak-gradient definition is well posed because its right-hand side is a bounded linear functional on H(div, K).
- When weak-function components come from u ∈H1(K), the weak gradient agrees with the classical gradient.
- The discrete weak gradient ∇d,r is the unique polynomial approximation in a chosen subspace V(K, r) of H(div, K).
- Weak gradients allow derivatives of functions without continuity across triangle boundaries, supporting heterogeneous approximation spaces.
- Analogous weak divergence and curl operators are proposed for future numerical applications.
4. A Weak Galerkin Finite Element Method.
The weak Galerkin method replaces H1(Ω) and the classical gradient with discrete weak-function spaces and discrete weak gradients on a triangular mesh.
- The method follows two principles: discretize H1(Ω) with element-and-boundary weak functions and replace ∇ with ∇d,r.
- On each element, discrete weak functions use degree-j interior polynomials and degree-ℓ boundary polynomials.
- Patching element spaces produces the global weak finite element space Sh(j, ℓ).
- The homogeneous-boundary subspace imposes vanishing boundary components on ∂Ω.
- The discrete weak gradient on each element is defined through the corresponding polynomial-space relation.
- The construction leaves V(T,r) unspecified beyond being a vector-valued polynomial subspace of degree at most r.
- The weak Galerkin approximation seeks uh={u0, ub} in Sh(j,ℓ), with ub set to the projected boundary data Qbg.
5. Examples of Weak Galerkin Method with Properties.
The section identifies properties needed for discrete weak gradients and presents two weak finite element spaces satisfying the numerical scheme's requirements. It also establishes that the discrete gradient reproduces projected classical gradients.
- The discrete gradient should vanish only for constant weak functions and should approximate the classical gradient after projection.These are the stated properties P1 and P2 guiding suitable weak finite element spaces.
- Two example spaces are proposed: Sh(j, j+1) with vector polynomials of degree j+1, and Sh(j, j) using a Raviart-Thomas-type vector space.The first uses interior degree j and boundary degree j+1; the second uses degree j for both components.
- For Sh(j, j+1), the discrete gradient vanishes if and only if both weak-function components equal the same constant.This verifies property P1 for the first example.
- The identity ∇d,j+1(Qhu) = Rh(∇u) shows that the discrete gradient of the projected function equals the L2 projection of the classical gradient.This verifies the approximation property P2 for smooth u.
- The projected discrete gradient is therefore identified as an excellent approximation of the classical gradient for functions in H1(T).
6. Mass Conservation of Weak Galerkin.
The weak Galerkin approximation retains the model problem's mass-conservation property through a numerical flux. The flux is also continuous across element edges.
- The weak Galerkin method retains the mass-conservation property with a numerical flux qh.The result is obtained by testing with an elementwise constant interior function and zero boundary component.
- The method's numerical flux qh · n is obtained from the discrete weak formulation and satisfies the elementwise conservation relation.
- The numerical flux qh · n is continuous across each element edge.This follows by choosing a test function with zero interior component and arbitrary boundary component.
7. Existence and Uniqueness for Weak Galerkin Approximations.
The section proves existence and uniqueness of weak Galerkin approximations under a dual H1+s-regularity assumption and sufficiently small mesh size. The analysis uses stability estimates, projection properties, and a duality argument.
- The weak Galerkin approximation is considered in the spaces Sh(j, j+1) and Sh(j, j).
- A Gårding-type inequality provides the stability estimate needed for the existence and uniqueness analysis.The bilinear form admits constants K and α1 in the stated estimate.
- The projection Πh is chosen so that its normal component is continuous across interior edges, supporting the discrete analysis.
- The uniqueness proof uses a duality approach based on the dual problem with homogeneous Dirichlet boundary conditions.The dual solution is assumed to have H1+s regularity for s in (0, 1].
- For sufficiently small but fixed mesh size h, the weak Galerkin method has a unique solution in both stated finite element spaces.
8. Error Analysis.
The error analysis compares the weak Galerkin approximation with an L2 projection through an error equation, then uses duality and projection estimates to derive L2 and discrete H1 error bounds. Under stated regularity and sufficiently small meshsize, the analysis also identifies superconvergence.
- Error equation: The analysis defines an error equation by comparing uh with the locally defined L2 projection Qhu in the weak finite element space.Qhu consists of local projections Q0u onto element polynomials and Qbu onto boundary polynomials.
- Discrete H1 estimate: Lemma 8.1 bounds the difference eh = uh − Qhu between the weak Galerkin approximation and the projected exact solution.The estimate is obtained from the error equation using eh as the test function, Cauchy–Schwarz, and Gårding’s inequality.
- L2 estimate: The L2 error analysis uses a dual problem with H1+s regularity and a standard duality argument.The dual problem has right-hand side Q0u − u0, and its regularity supports the subsequent estimates.
- Discrete H1 estimate: Theorem 8.3 derives a discrete H1-norm error estimate when u ∈ Hm+1(Ω) with 0 ≤ m ≤ j + 1.The estimate assumes the dual problem’s regularity from Lemma 8.2.
- L2 estimate: Theorem 8.4 provides a further error estimate under Theorem 8.3’s assumptions, with sufficiently small mesh-size.The analysis further specializes the result for Hj+2 regularity and discusses the case of full H2 dual regularity.
- Superconvergence: For polynomial order j ≥ 0 on each triangular element, estimate (8.13) reveals superconvergence of the weak Galerkin approximation.The superconvergence conclusion is tied to the elementwise polynomial structure and the preceding error estimate.