[Paper Review] A weak Galerkin finite element method for Burgers' equation

Yanli Chen, Tie Zhang|arXiv (Cornell University)|Jul 19, 2016

Advanced Numerical Methods in Computational Mathematics12 references3 citations

TL;DR

This paper proposes a weak Galerkin (WG) finite element method for solving the one-dimensional Burgers' equation using a novel weak variational formulation. The method establishes both semi-discrete and fully-discrete schemes, proving optimal convergence rates of order $k+1$ in the discrete $H^1$-norm and $L^2$-norm for polynomial degree $k$, validated by numerical experiments with exact solutions and convergence studies.

ABSTRACT

We propose a weak Galerkin(WG) finite element method for solving the one-dimensional Burgers' equation. Based on a new weak variational form, both semi-discrete and fully-discrete WG finite element schemes are established and analyzed. We prove the existence of the discrete solution and derive the optimal order error estimates in the discrete $H^1$-norm and $L^2$-norm, respectively. Numerical experiments are presented to illustrate our theoretical analysis.

Motivation & Objective

• To develop a stable and accurate numerical scheme for solving Burgers' equation, a simplified model of the Navier-Stokes equations.
• To address the challenge of solving Burgers' equation, especially in the high-Reynolds-number (nearly hyperbolic) regime.
• To establish a weak Galerkin finite element method with a new weak variational form that ensures positive definiteness of the bilinear form.
• To derive optimal error estimates in both discrete $H^1$-norm and $L^2$-norm for the semi-discrete and fully-discrete schemes.
• To validate the theoretical findings through numerical experiments with exact solutions and convergence analysis.

Proposed method

• A new weak variational formulation is derived by integrating by parts and symmetrizing the nonlinear convective term, leading to a positive-definite bilinear form.
• The discrete weak derivative is defined in a local finite element space using orthogonal polynomials of degree $k$, ensuring local conservation and stability.
• The semi-discrete WG scheme is formulated in space using $H^1$-conforming finite element spaces with weak functions $v = \{v^0, v^a, v^b\}$, where $v^0$ is the interior value and $v^a, v^b$ are the values at element boundaries.
• The fully-discrete scheme is constructed using backward Euler time discretization, leading to a linear system at each time step.
• The discrete weak derivative satisfies a discrete integration-by-parts identity, crucial for stability and error analysis.
• Error estimates are derived using projection-based analysis and approximation properties of the weak Galerkin spaces.

Experimental results

Research questions

• RQ1 Can a weak Galerkin finite element method be effectively applied to the one-dimensional Burgers' equation with optimal convergence rates?
• RQ2 Does the proposed weak variational formulation ensure the positive definiteness of the bilinear form, enabling stable numerical solutions?
• RQ3 What is the convergence behavior of the WG finite element method in both $H^1$-norm and $L^2$-norm for different polynomial degrees $k$?
• RQ4 How do the numerical solutions compare with exact solutions for varying viscosity $\nu$ and mesh refinement?
• RQ5 Can the method maintain high accuracy even in the limit of small viscosity, approaching the hyperbolic regime?

Key findings

• The proposed WG finite element method achieves optimal convergence order $k+1$ in the discrete $H^1$-norm for polynomial degree $k$.
• Optimal convergence order $k+1$ is also achieved in the $L^2$-norm, as confirmed by theoretical error estimates.
• Numerical experiments with $\nu = 0.1$ and $\nu = 0.01$ show excellent agreement between numerical and exact solutions at $t_n = 0.1, 0.4, 0.6, 0.8, 1.0$.
• Convergence plots for $k=0$ and $k=1$ at $t_n = 1.0$ confirm $k+1$ order convergence in both $H^1$ and $L^2$ norms across $\nu = 0.1, 0.01, 0.001$.
• The method remains stable and accurate even for small viscosity values, such as $\nu = 0.001$, demonstrating robustness in the nearly hyperbolic regime.
• The numerical results validate the theoretical error estimates and confirm the effectiveness of the proposed weak variational formulation.