A weak Galerkin finite element method for Burgers’ equation

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 fullydiscrete 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 1norm and L 2 -norm, respectively. Numerical experiments are presented to illustrate our theoretical analysis. Keywords: weak Galerkin finite element method; Burgers' equation; optimal error estimates.where I = (0, 1), J = (0, T ] with T < ∞, u is the unknown velocity, ν = 1 Re , Re is the Reynolds number and g(x) is a given function.Burgers' equation (1a) has been widely used for various applications, such as modeling of gas dynamics and turbulence [1], describing wave propagation in acoustics and

Weak Galerkin finite-element method for time-fractional nonlinear integro-differential equations

“…And (4.30), (4.31) show that Q h u is a good approximation for function in H 1 0 (I ) ∩ H s+1 (I ), s ≥ 0. To simplify the proof, we introduce the following projection function, which can be found in Chen and Zhang (2019).…”

“…Zhang and Tang (2016) constructed a WG finite-element scheme for elliptic equation in one-dimensional case. Chen and Zhang (2019) proposed the WG finite-element scheme for Burgers equation. Zhou et al (2018) formulated the multi-term time-fractional diffusion equations using this method.…”

Weak Galerkin and Weak Group Finite Element Methods for 2-D Burgers' Problem

“…Experiments with the GrFEM have shown an increase in economy and in the nodal accuracy compared to FE solutions of the Burgers' equations [3,5,6] and for other problems [8,4,9,27,32] . Recently, the weak Galerkin finite element method has catched much consideration in the field of numericaln PDEs , The idea of the weak Galerkin method was first introduced by Wang and Ye [14], the method was applied to the second order elliptic equations [1,15,16,19,20,22,24,28], the Stokes equations [17,21], Parabolic equations [10,11,25], biharmonic equations [23,26], Navier-Stokes equations [7,18,30] and 1-D Burgers' equation [31], etc. In this paper we present (WGFEM) and (WGrFEM) for 2-D Burgers' problem with a fully-discrete approximation for the time variable, the backward-difference formula for the time variable is examined and the stability and error estimate are proved for these methods.…”

Weak Galerkin finite element method for a class of time fractional generalized Burgers' equation

“…In [22], Lin introduced a new WG finite element method to solve convection-diffusion-reaction equations in the convection dominated regime. Although the WG finite element method has been employed for the viscous Burgers' equation [23]. To the best of our knowledge, time fractional generalized Burgers' equation has not been studied by this method.…”