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Weak Galerkin Finite Element Methods for Parabolic Equations

Qiaoluan H. Li, Junping Wang

arXiv:1212.3637v2math.NAcs.NA

TL;DR

For parabolic equations, the paper develops weak Galerkin finite element schemes using a discrete weak gradient and allowing totally discontinuous approximation functions. Numerical tests report optimal convergence rates in both L2 and weak Galerkin H1 norms for a smooth exact solution with Dirichlet data.

  • Problem

    The paper addresses the numerical solution of parabolic initial-boundary value problems.

    Source: §1 · p. 1

  • Method

    The weak Galerkin approach uses a discrete weak gradient and develops continuous-time and backward-Euler schemes.

    Source: §1 · p. 2, §4.1 · p. 5, §4.2 · p. 9

  • Results

    For smooth exact solutions with Dirichlet boundary data, numerical tests observe optimal convergence rates in both L2 and weak Galerkin H1 norms.

    Source: §5 · p. 16

  • Takeaways & Limitations

    The method permits totally discontinuous approximation functions while preserving the energy conservation law.

    Source: §1 · p. 2

    The stated setting uses polygonal or polyhedral Lipschitz domains in dimensions 2 or 3 and a symmetric, bounded coefficient matrix.

    Source: §1 · p. 1

Abstract

from arXiv · show

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal order error estimates in both H^1 and L^2 norms are established. Numerical tests are performed and reported.

1. Introduction

The paper introduces a weak Galerkin finite element method for parabolic equations, building on a discrete weak gradient. It outlines continuous- and discrete-time schemes, energy conservation, error analysis, and numerical experiments.

1. Introduction

  • Earlier numerical approaches discussed include finite element, discontinuous Galerkin, and finite volume methods, with the latter enforcing an integral conservation law on mesh-defined control volumes.The cited approaches are –.Source: §1 · p. 2
  • The proposed weak Galerkin method uses a discrete weak gradient, allows totally discontinuous approximation functions, and preserves the energy conservation law.The discrete weak gradient operator was proposed in [7].Source: §1 · p. 2
  • The paper develops continuous- and discontinuous-time schemes, proves energy conservation and optimal-order H1 and L2 error estimates, and reports numerical experiments.Source: §1 · p. 2

2. The Weak Galerkin Method

The method replaces the classical gradient with a discrete weak gradient in a Galerkin formulation. It defines the finite element spaces and develops both continuous-time and backward-Euler discrete-time schemes.

2. The Weak Galerkin Method

  • The weak Galerkin space pairs element-interior polynomials with edge polynomials, and its subspace for homogeneous boundary values sets boundary-edge values to zero.Source: §2 · p. 2
  • The analysis uses local L2 projections for scalar and vector spaces and assumes an H(div)-conforming projection with a stated approximation property.Source: §2 · p. 2
  • The space construction criteria require zero discrete weak gradient to imply a constant function and require projected gradients to approximate the exact gradient.The approximation criterion gives an error bound proportional to h^m for functions with the stated regularity.Source: §2 · p. 3
  • The method uses a discrete weak gradient in place of the classical gradient in its trial-and-test Galerkin formulation.The discrete weak gradient is defined elementwise, with the outward normal used on each element boundary.Source: §2 · pp. 2–3
  • The continuous-time scheme uses the weak bilinear form, while the discrete-time scheme replaces the time derivative with a backward difference quotient.The discrete scheme is given at time levels t_n=nk.Source: §2 · p. 3

3. Energy Conservation of Weak Galerkin

The section establishes that the weak Galerkin approximation retains an integral energy conservation property. Its local numerical flux is continuous across element edges.

3. Energy Conservation of Weak Galerkin

  • The balance describes control-volume energy change over time in relation to boundary heat flux and source energy.The passage identifies q = −a∇u as the heat-flow rate.Source: §3 · p. 4
  • The weak Galerkin approximation retains the integral energy conservation property for the parabolic problem.The proof applies a test function localized to a control volume and uses the weak gradient and projection definitions.Source: §3 · p. 4
  • The numerical flux has a continuous normal component across each element edge.The paper verifies this by choosing a test function with zero interior component and arbitrary edge component.Source: §3 · p. 4

4. Error Analysis

The section analyzes errors in continuous- and discrete-time weak Galerkin schemes relative to projections of the exact solution, using an elliptic projection to derive optimal-order estimates. Under the assumptions of Theorem 4.5 and with a time-independent coefficient matrix, it also establishes a discrete-gradient error estimate with an h^{2r}-scaled regularity term.

4.1. Continuous Time Weak Galerkin Finite Element Method.

  • Under u ∈ H1+r(Ω), Theorem 4.1 establishes an error estimate for the continuous-time approximation uh − Qhu.The analysis derives an error equation and uses coercivity and approximation estimates to bound the error.Source: §4.1 · p. 5

4.2. Discrete Time Weak Galerkin Finite Element Method.

  • The discrete-time analysis establishes a Poincaré-type inequality for the weak gradient and error estimates for both en and ∇den.The estimates concern the backward Euler approximation under u ∈ H1+r(Ω).Source: §4.2 · pp. 7, 9, 10, 11

4.3. Optimal Order of Error Estimation in L2.

  • The L2 analysis splits uh − Qhu into θ = uh − Ehu and ρ = Ehu − Qhu, bounding the elliptic-projection error separately.The method applies elliptic-problem error estimates to ρ and focuses the parabolic analysis on θ.Source: §4.3 · p. 11
  • The continuous-time analysis gives an L2 estimate and an H1 estimate when the coefficient matrix a is independent of time.The estimates assume the regularity conditions in Theorems 4.1 and 4.3; the H1 estimate is stated in Theorem 4.4.Source: §4.3 · pp. 11–13
  • Theorem 4.5 establishes an error estimate for the backward Euler weak Galerkin approximation relative to the L2 projection.The error is decomposed into the elliptic-projection error and the remaining parabolic error.Source: §4.3 · pp. 13–14
  • The estimate holds under Theorem 4.5's assumptions when the coefficient matrix a is independent of time.Source: §4.3 · p. 15
  • Theorem 4.6 bounds the discrete gradient error at t_n using twice the initial discrete gradient error and an h^{2r}-scaled regularity term.The theorem's displayed bound begins ∥∇d(U^n−Q_hu(t_n))∥_2 ≤ 2∥∇d(U^0−Q_hu(0))∥_2 + Ch^{2r}(…); the supplied passage truncates the regularity terms.Source: §4.3 · p. 15
  • The proof estimates ∇dθ^n by testing with the backward difference of θ^n and repeating the resulting estimate.Source: §4.3 · p. 15

5. Numerical Experiments

Numerical tests cover smooth Dirichlet heat problems, mixed-boundary heat problems, and a full-tensor parabolic problem. They show optimal convergence in the tested norms, with the full-tensor example also exhibiting higher-than-predicted discrete H1 convergence.

Numerical setup

  • The numerical studies use a uniform triangular mesh, piecewise-constant interior and edge functions, and the lowest-order Raviart–Thomas element RT0.The discrete weak space is Sh(0, 0).Source: §5 · p. 15

Example 1

  • For smooth inhomogeneous Dirichlet heat data, tests with k=h and k=h² show optimal convergence in L2 and weak Galerkin H1, consistent with theory.With k=h, time-step error dominates; with k=h², mesh-size error dominates.Source: Table 1 · p. 16, Table 2 · p. 16, §5 · p. 16

Mixed boundary condition

  • For the mixed-boundary heat problem, the reported error achieves optimal convergence rates in all norms.The tests use Robin boundary conditions and include both k=h and k=h² settings.Source: §5 · p. 16, Table 3 · p. 17, Table 4 · p. 17

Example 2

  • For the full-tensor parabolic example, L2 convergence matches theoretical rates, while discrete H1 converges at O(h2), one order above prediction.The authors suggest this may indicate superconvergence between the weak Galerkin approximation and the L2 projection.Source: Table 5 · p. 17, §5 · p. 17, Table 6 · p. 18
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