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

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a b s t r a c tA covolume method is proposed for the mixed formulation of second-order elliptic problems. The solution domain is divided by a quadrilateral grid, corresponding to which a nonoverlapping dual grid is constructed. The velocity and pressure are approximated by the lowest-order Raviart-Thomas space on quadrilaterals. We prove its first order optimal rate of convergence for the approximate velocities in the H(div)-norm as well as for the approximate pressures in the L 2 -norm. A second order error estimate between a suitable projection of the exact velocity (or pressure) and the approximate velocity (or approximate pressure) is also presented. Numerical experiments are provided to illustrate the error behavior of the scheme.

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