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This paper is devoted to investigating the optimal convergence order of a weak Galerkin finite element approximation to a second-order parabolic equation whose solution has lower regularity. In many applications, the solution of a second-order parabolic equation has only H 1 + s smoothness with 0 < s < 1 , and the numerical experiments show that the weak Galerkin approximate solution exhibits an optimal convergence order of 1 + s . However, the standard numerical analysis for weak Galerkin finite element method always requires that the exact solution should have at least H 2 smoothness. Our work fills the gap in the error analysis of weak Galerkin finite element method under lower regularity condition, where we prove the convergence order is of 1 + s . The main strategy of analysis is to introduce an H 2 -regular finite element approximation to discretize the spatial variables in variational equation, and then we analyze the error between this semi-discretized solution and the full discretized weak Galerkin solution. Finally, we present some numerical experiments to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j + 1 with j ≥ 0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L 2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we apply a new weak Galerkin mixed finite element method (WGMFEM) with stabilization term to solve heat equations. This method allows the usage of totally discontinuous functions in the approximation space. The WGMFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. In addition, we develop and analyze the error estimates for both continuous and discontinuous time WGMFEM schemes. Optimal order error estimates in both L 2 and triple-bar ⫼ ⋅ ⫼ norms are established, respectively. Finally, numerical tests are conducted to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]