Optimal convergence analysis of weak Galerkin finite element methods for parabolic equations with lower regularity

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 \(\varvec{H}^{\varvec{1+s}}\) ${\mathit{H}}^{\mathbf{1}\mathbf{+}\mathit{s}}$ smoothness with \(\varvec{0<s<1}\) $\mathbf{0}\mathbf{<}\mathit{s}\mathbf{<}\mathbf{1}$ , and the numerical experiments show that the weak Galerkin approximate solution exhibits an optimal convergence order of \(\varvec{1+s}\) $\mathbf{1}\mathbf{+}\mathit{s}$ . However, the standard numerical analysis for weak Galerkin finite element method always requires that the exact solution should have at least \(\varvec{H}^{\varvec{2}}\) ${\mathit{H}}^{\mathbf{2}}$ 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 \(\varvec{1+s}\) $\mathbf{1}\mathbf{+}\mathit{s}$ . The main strategy of analysis is to introduce an \(\varvec{H}^{\varvec{2}}\) ${\mathit{H}}^{\mathbf{2}}$ -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.